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| 8 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here is some recommended reading before getting started with this lesson.
Consider a circle with center O and point P outside of the circle. Using a straightedge and compass, can you construct a tangent to the circle through the given point?
Given a circle and a point outside the circle, a compass and straightedge can be used to draw a tangent line from the point to the circle.
The Inscribed Right Triangle Theorem can be used to justify why this construction works.
Consider OA, a radius of ⊙O.
In circle M, ∠OAP is an inscribed angle on a diameter of ⊙M. Since an inscribed angle opposite the diameter is a right angle, ∠OAP is a right angle.
As can bee seen, OA⊥PA. The radius of ⊙O is perpendicular to a line that passes through a point on the circle. Therefore, by the Tangent to Circle Theorem, PA is tangent to the circle.
See solution.
How can a tangent line from a point outside of the given circle be constructed?
Since point A is a point outside ⊙O, B should be the point of tangency in order for AB to be tangent to the circle. On the example shape, by extending AB, it can be observed that B is the point of tangency.
Constructing a tangent from an outer point will help locate the point of tangency for a tangent drawn from A. Recall the steps in constructing a tangent.
The lines of symmetry of a circle are the lines that passes through the center of the circle.
Suppose that a tangent line drawn from an outer point intersects a circle at A. When A is reflected across the line that passes through P and O, its image will also be a point of tangency for another tangent.
Two tangent segments drawn from a common external point to the same circle are congruent.
If AB and AC are tangent segments to ⊙O, then AB≅AC.
Consider two triangles.
These two triangles can be visualized in the diagram.
Note that B and C are points of tangency. Therefore, by the Tangent to Circle Theorem, ∠B and ∠C are right angles. Consequently, △ABO and △ACO are right triangles.
Because all radii of the same circle are congruent, it can be said that OB and OC are congruent. Moreover, △ABO and △ACO share the same hypotenuse OA. By the Reflexive Property of Congruence, OA is congruent to itself.
Combining all of this information, it can be said that the hypotenuse and one leg of △ABO are congruent to the hypotenuse and the corresponding leg of △ACO.
Therefore, by the Hypotenuse-Leg Theorem, △ABO and △ACO are congruent triangles. Since corresponding parts of congruent figures are congruent, it can be said that AB and AC are congruent.
AB≅AC
In the diagram, all three segments are tangent to circle P.
The points D, E, and F are the points where the segments touch the circle. If AB=12, BC=10, and CA=6, find AE.
Use the External Tangent Congruence Theorem.
From the graph, it can be seen that AD and AE are tangent segments with a common endpoint outside ⊙P. By the External Tangent Congruence Theorem, AE and AD are congruent.
Remove parentheses
Add and subtract fractions
LHS/2=RHS/2
In a factory, two gears are connected by a rubber band such that BC becomes a common tangent to both gears.
Let's illustrate the situation. Notice that BC is tangent to both circles which means they are perpendicular to the radii of the circles. Let's label the segment between the gears' centers AD. Additionally, BA and CD are parallel segments.
Starting at point B, let's draw a segment to a point on CD in such a way that the segment is parallel to AD. This results in a right triangle with legs of 20 and 3 inches, respectively, and a parallelogram with two sides that are 1 inches long.
If we find the length of the segment that is parallel to AD, we also find the length of AD. To do that, we will use the Pythagorean Theorem.
The length of the parallel line, and therefore AD, is sqrt(409) inches.
Find the values of x and y.
According to the External Tangent Congruence Theorem, tangents drawn from a common external point are congruent segments. That means PA and PC are congruent. Let's show this relationship in the diagram.
Similarly, PB and PD are congruent. Notice that PB is the sum of PA and AB. At the same time, PD is the sum of PC and CD. Let's show these relationships in the diagram.
Now we can write two equations. &3x+3=y+8 & (3x+3)+(4y-29)=(y+8)+(x+6) If we combine these, we get a system of equations which we will solve by using the Substitution Method.
Having solved the system for x, we can substitute this into Equation (I) to calculate the value of y.
The line QR is tangent to the circles ⊙P and ⊙O. What is the distance between the tangent points Q and R if ⊙P and ⊙O have radii of 14 and 8 feet respectively and the length of AB is 17 feet? Give the answer in exact form.
Let's add all of the given information to the diagram. Notice that PQ and PA are both radii of ⊙ P. Also, OB and OR are both radii of ⊙ O.
By the Tangent to Circle Theorem, QR is perpendicular to both PQ and OR. To determine the length of QR, we can from O draw a parallel segment to QR, which intersects PQ. This segment will be congruent to QR.
Examining the diagram, we can spot a right triangle where the hypotenuse and the shorter leg are known.
Using the Pythagorean Theorem, we can find the measure of the unknown leg and thereby also QR.
The length of the horizontal leg is 3sqrt(165) feet. That is also the length of QR.
What is m∠ADC if BP is tangent to the circle?
We know that BP is tangent to ⊙ A. According to the Tangent to Circle Theorem, ∠ ABP is a right angle.
According to the Interior Angles Theorem, the angle measures of △ ABP add to 180^(∘). With this information, we can write and solve an equation containing the central angle ∠ BAP.
Notice that ∠ BAP and ∠ CAD are vertical angles. According to the Vertical Angles Theorem, these angles are congruent. Also, notice that AC and AD are both radii of the circle which means these segments are congruent. Therefore, according to the Isosceles Triangle Theorem, the base angles of △ ACD are congruent.
Now we can determine the measure of ∠ ADC.