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Similarity, Proof, and Trigonometry

Similarity Theorems About Triangles

In this lesson, the similarity of triangles will be used to prove some claims about triangles.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

Which of the following conditions guarantee that two triangles are similar?


Challenge

Solving Problems Using Triangle Similarity

The street lamps are and feet tall. How tall is the Grim Reaper?

Doubleshadow.jpg

Explore

Investigating Triangle Proportionality

Move the point on the side of the triangle. The applet draws a line parallel to another side of the triangle and gives the length of four line segments.

  • Move the vertices and investigate the relationship between these lengths. What do you notice?
  • Explore different triangles in relation to the pyramids!

Discussion

Triangle Proportionality Theorem

The previous exploration can lead to discovering the following claim, which is often referred to as the Side-Splitter Theorem.

If a segment parallel to one of the sides of a triangle is drawn between the other sides, the segment divides the other two sides proportionally.

Triangle ABC with segment DE drawn parallel to AB

Based on the diagram, the following relation holds true.

If then

Proof

Since and are parallel, by the Corresponding Angles Theorem, and are congruent. Similarly, and are congruent.

Triangle ABC with segment DE drawn parallel to AB
Therefore, by the Angle-Angle Similarity Theorem, and are similar. Consequently, their corresponding sides are proportional. Applying the Segment Addition Postulate, both numerators can be rewritten. Substituting these expressions into the equation above, the required proportion will be obtained.

Pop Quiz

Practice Using the Triangle Proportionality Theorem

Example

Using the Triangle Proportionality Theorem to Draw Segments

In the following example the Triangle Proportionality Theorem can be used after rearranging the segments to form triangles. Given the segments on the diagram, construct a segment of length

Three segments of length 1, a, and b.

Answer

Four segments of length 1, a, b, and ab.

Hint

Use that

Solution

Move the slider to rearrange the given line segments. Note that this can also be done on paper using a straightedge to draw straight lines, followed by using a compass to copy the segments.

Label the points on this rearranged graph, connect the endpoints of the segments of length and and draw a parallel line to this connecting line through the endpoint of the segment of length

In the transversal is parallel to the side According to the Triangle Proportionality Theorem, this transversal cuts sides and proportionally. Substituting the length of and gives an equation which can be solved for the length of

This construction gave a segment of length

Discussion

Converse Triangle Proportionality Theorem

The converse of the Side-Splitter Theorem is also true.

If a segment is drawn between two sides of a triangle such that it divides the sides proportionally, the segment is parallel to the third side in the triangle.

Based on the diagram, the following relation holds true.

If then

Proof

The given proportion can be rearranged to get the proportionality of two sides of and
Rewrite

Segment Addition Postulate

This means that is a dilation of from point with scale factor A dilation moves a segment to a parallel segment, so the proof is complete.

If then

Example

Solving Problems With the Converse Triangle Proportionality Theorem

Show that is a parallelogram.

Quadrilateral ABCD with points P,Q,R,S on the four sides. The given distances are AP=4.2, PB=1.8, BQ= 2.4, QC=5.6, CR=6.3, RD=2.7, DS=2.1 SA=4.8.

Hint

Draw the diagonals of quadrilateral

Solution

Draw diagonal of quadrilateral and focus on the two triangles and

The given measures of the segments make it possible to derive the ratios according to how the transversal divides sides and of It can be seen that the two ratios are equal. Therefore, according to the converse of the Triangle Proportionality Theorem, the transversal is parallel to the diagonal A similar calculation shows that cuts the sides of proportionally. Therefore, is also parallel to the diagonal Since and are both parallel to they are also parallel to each other.

After drawing the diagonal the previously found proportions also show that cuts the sides and of proportionally. Additionally, cuts the sides and of proportionally. This implies that and are both parallel to Therefore, they are also parallel to each other.

This completes the proof that opposite sides of quadrilateral are parallel. Hence, by definition, it is a parallelogram.

Quadrilateral ABCD with points P,Q,R,S on the four sides. The given distances are AP=4.2, PB=1.8, BQ= 2.4, QC=5.6, CR=6.3, RD=2.7, DS=2.1 SA=4.8.

Discussion

Three Parallel Lines Theorem

The following theorem is a corollary of the Side-Splitter Theorem.

If three parallel lines intersect two transversals, then they divide the transversals proportionally.

three parallel lines and two transversals

Applying the theorem to the diagram above, the following proportion can be written.

Proof

In the diagram, draw and let be the point of intersection between this segment and line

three parallel lines and two transversals
Next, separate and
three parallel lines and two transversals
Since by the Triangle Proportionality Theorem divides and proportionally. Applying the same reasoning in it can be said that divides and proportionally. Finally, the Transitive Property of Equality leads to the desired proportion, showing that the three parallel lines divide the transversals proportionally.

Pop Quiz

Practice Using the Three Parallel Lines Theorem

Example

Solving Problems Using the Three Parallel Lines Theorem

Construct points that divide the given segment into five congruent pieces.

A segment with endpoints A and B.

Hint

Draw a different segment and extend it with four congruent copies.

Solution

Draw a ray starting at and use a compass to copy any length five times on this ray. This gives five points, and

Connect with the last point, and construct parallel lines to this segment through the other points. Mark the intersection points of these lines with segment

According to the Three Parallel Lines Theorem, these transversals divide segments and proportionally. Since, by construction, the segments on have equal length, this means that points and divide into congruent segments.

Explore

Investigating Inner Triangles of a Triangle

The examples until now were based on similar triangles generated by parallel lines. Is it possible to cut a triangle to two similar triangles using a line starting from a vertex?

Move the vertices and the point on the side of the triangle. It is possible to find an arrangement when the two inner triangles are similar to each other and to the original triangle. Find such a diagram.

Discussion

Right Triangle Similarity Theorem

The previous exploration can lead to the following claim.

Given a right triangle, if an altitude is drawn from the vertex of the right angle to the hypotenuse, then the two triangles formed are similar to the original triangle and to each other.

A right triangle with the altitude to the hypotenuse drawn

According to this theorem, there are three relations that hold true for the diagram above.

Proof

Start by separating the two triangles formed by the altitude from
A right triangle with the altitude to the hypotenuse drawn
By the Reflexive Property of Congruence, and Also, since all right angles are congruent, it is obtained that and
and and

Applying the Angle-Angle (AA) Similarity Theorem, it can be concluded that and are similar and and are similar. Then, by the Transitive Property of Congruence, and are also similar.

and

Discussion

Corollaries of the Right Triangle Similarity Theorem

The following two claims are corollaries of the Right Triangle Similarity Theorem

Rule

Geometric Mean Altitude Theorem

Given a right triangle, if an altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of this altitude is the geometric mean between the measures of the two segments formed on the hypotenuse.

Right triangle with the altitude from the right angle drawn

Based on the diagram above and by definition of the geometric mean, the following relation holds true.

or

The Geometric Mean Altitude Theorem is also known as the Right Triangle Altitude Theorem and the Geometric Mean Theorem.

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude are similar. Then, by definition of similar triangles, the lengths of corresponding sides are proportional.

Applying the Properties of Equality, this proportion can be rewritten without fractions.

Rule

Geometric Mean Leg Theorem

Given a right triangle, if the altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of each leg of the triangle is the geometric mean between the length of the hypotenuse and the length of the segment formed on the hypotenuse adjacent to the leg.

Right triangle with the altitude from the right angle drawn

Based on the diagram above, the following relations hold true.

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude are similar to Then, by definition of similar triangles, the length of corresponding sides are proportional.

Applying the Properties of Equality, the proportion above can be rewritten without fractions.

Example

Solving Problems With the Geometric Mean Leg Theorem

Represented on the figure is a right triangle and an altitude .

Triangle ABC with altitude CD starting from the right angle at C. The given measurements are CB=5cm and AD=6cm.

Using the given measurements, find the length of

Hint

First, find the length of

Solution

The answer can be found in two steps. The first step is to find the length of

Finding

The Geometric Mean Leg Theorem shows the connection between the length of and

The length of can be expressed using the length of

The relationship AB=28.8+DB added
By substituting these expressions in the equation given by the Geometric Mean Leg Theorem, the results can be expressed in an equation that can then be solved for the length of
Rewrite
Evaluate right-hand side
This gives two solutions, and Since represents the length of segment only the positive solution is meaningful, as a length of a segment can not be negative.

Finding

This gives the length of the other segment formed by the altitude in the right triangle

The measure DB=5cm added

The Geometric Mean Altitude Theorem gives a connection between the length of and

This relationship can be used to find the length of the altitude.
Solve for
The length of the altitude is centimeters.

Pop Quiz

Practice the Geometric Mean Leg Theorem

Discussion

Pythagorean Theorem

In a right triangle, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs.

Right triangle

Proof

Using Similarity

First, the altitude from the right angle to the hypotenuse will be drawn. This will divide the hypotenuse into two segments.

Right triangle and height from the right angle

By applying the Geometric Mean Leg Theorem, the lengths of the legs can be related to the length of the hypotenuse. Next, these two equations will be added and factored out on the right-hand side. From construction, is equal to After substituting this into the equation above and simplifying, the Pythagorean Theorem is obtained.

Closure

Solving Problems Using Triangle Similarity

To conclude this lesson, the opening challenge will be revisited. The challenge shows a diagram consisting of the Grim Reaper and two street lamps at and feet tall. How tall is the Grim Reaper?

Doubleshadow.jpg

Hint

The lamps, the head of the Grim Reaper, and the shadows of the Grim Reaper's head are on a straight line.

Solution

The lamps and the figure stand vertically. Hence, they can be represented by parallel segments and . Notice that the shadows just reach the lampposts. Taking a look at the shadow touching the taller post, it can be derived that the lamppost bottom the head of the Grim Reaper and the lamppost top are on a straight line. Applying this same logic to the other shadow implies that and are also on a straight line.

Segment splits both and It is also parallel to one side of both triangles. That means the combination of two dilations maps to

  • A dilation with center and scale factor maps to
  • A dilation with center and scale factor maps to
The scale factor of this combined similarity transformation is the product of the two scale factors.
Continuing, notice that Point is between and Therefore, the Segment Addition Postulate guarantees that Then, to divide this equality by and to rearrange it gives a relationship between the scale factors of the two dilations.
Rewrite

The product of the two scale factors is the scale factor of the similarity transformation that maps the foot lamppost to the foot lamppost. Hence, the scale factor is An equation can now be written and solved to find the individual scale factors.
Solve for
This finding, is the scale factor of the dilation that maps to Since the length of can now be calculated. Segment represents the Grim Reaper, who is now known to stand at feet tall.
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