Big Ideas Math: Modeling Real Life, Grade 8
BI
Big Ideas Math: Modeling Real Life, Grade 8 View details
6. The Converse of the Pythagorean Theorem
Continue to next subchapter

Exercise 24 Page 414

Calculate the distance between the given points and use the Converse of the Pythagorean Theorem.

No.

Practice makes perfect

To determine whether the given points form a right triangle, we will start by plotting and labeling the points. Then, we will connect them on a coordinate plane.

triangle

Now that we can see the triangle, we will need to measure the side lengths and then use the Converse of the Pythagorean Theorem to check whether the lengths create a right triangle.

Measuring the Side Lengths

First, let's measure the horizontal and vertical distances between the points A and C.
triangle
Knowing the horizontal and vertical distances, we are able to find the distance d_(AC) between A and C by using the Pythagorean Theorem. Keep in mind that segment AC is the hypotenuse of the triangle that we created, so its value must be substituted for c in the theorem.
a^2+b^2=c^2
8^2+ 19^2=( d_(AC))^2
â–Ľ
Solve for d_(AC)
64+361=(d_(AC))^2
425=(d_(AC))^2
sqrt(425)=d_(AC)
d_(AC)=sqrt(425)
The distance between A and C is sqrt(425). Let's now measure the horizontal and vertical distance between C and B as well as between B and A.
triangle

With these values, we can calculate the distances.

Distance Pythagorean Theorem Solve
Between C and B 7^2+ 22^2=( d_(CB))^2 d_(CB)= sqrt(533)
Between B and A 15^2+ 3^2=( d_(BA))^2 d_(BA)= sqrt(234)

Using the Converse of the Pythagorean Theorem

Finally, let's recall the Converse of the Pythagorean Theorem.

Converse of the Pythagorean Theorem

If the Pythagorean Theorem is true for the side lengths of a triangle, then the triangle is a right triangle.

We will use this to determine whether our triangle, which has the sides lengths sqrt(425), sqrt(533), and sqrt(234), is a right triangle. It is important to make sure that the longest side length is substituted as c in this equation or it will not work properly!
a^2+b^2=c^c
( sqrt(425))^2+( sqrt(234))^2? = ( sqrt(533))^2
â–Ľ
Simplify
425+234? = 533
659 ≠ 533 *
The Pythagorean Theorem proved false for the obtained side lengths. Therefore, we can conclude that the triangle formed by the given points is not a right triangle.