Exercise 36 Page 419

The first thing we will do is plot the points and connect them on a coordinate plane.

Now we can see the triangle. We need to measure the side lengths and use the Converse of the Pythagorean Theorem to tell whether the points form a right triangle.

Measuring the Side Lengths

We will use the Pythagorean Theorem to calculate the distance between A and B. The horizontal and vertical distances are the legs of the triangle and the distance between A and B is the hypotenuse.

a^2+b^2=c^2

SubstituteValues

Substitute values

3^2+ 2^2=c^2

▼

Solve for c

CalcPow

Calculate power

9+4=c^2

AddTerms

Add terms

13=c^2

RearrangeEqn

Rearrange equation

c^2=13

SqrtEqn

sqrt(LHS)=sqrt(RHS)

c=sqrt(13)

The distance between A and B is sqrt(13). Let's now measure the horizontal and vertical distance between A and C as well as between B and C.

We can calculate the distances between these pairs of points in the same way.

Distance	Pythagorean Theorem	Solve
Between A and C	2^2+ 3^2=c^2	c= sqrt(13)
Between B and C	5^2+ 1^2=c^2	c= sqrt(26)

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 use this to determine whether our triangle with the sides lengths sqrt(13), sqrt(13), and sqrt(26) is a right triangle. It is important to substitute the longest side length as c in this equation!

a^2+b^2=c^c

SubstituteValues

Substitute values

sqrt(13)^2+ sqrt(13)^2? = sqrt(26)^2

▼

Simplify

PowSqrt

( sqrt(a) )^2 = a

13+13? =26

AddTerms

Add terms

26=26 ✓

The Pythagorean Theorem proves that the points form a right triangle.