body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

6. Secants, Tangents, and Angle Measures

Continue to next subchapter

Exercise 51 Page 767

Find the length of each side to show that the triangle is a right triangle.

m∠ X ≈ 51.3^(∘)

Practice makes perfect

Let's start by drawing △ XYZ whose vertices are the given ones.

Before finding m ∠ X, let's find the length of each side by using the Distance Formula. d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)First, we will find XZ.

XZ = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)

SubstitutePoints

Substitute ( 2,2) & ( 7,-2)

XZ = sqrt(( 7- 2)^2 + ( -2-2)^2)

▼

Simplify right-hand side

Subtract terms

XZ = sqrt(5^2 +(-4)^2)

Calculate power

XZ = sqrt(25 + 16)

Add terms

XZ = sqrt(41)

The computations to find the other two side lengths are summarized in the following table.

Points	d = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)	Length
X( 2, 2) and Y( 2, -2)	XY = sqrt(( 2- 2)^2 + ( -2- 2)^2)	XY = 4
Y( 2, -2) and Z( 7, -2)	YZ = sqrt(( 7- 2)^2 + ( -2-( -2))^2)	YZ = 5

Since XY^2+YZ^2 = XZ^2, we conclude that △ XYZ is a right triangle with ∠ Y being the right angle.

Next, applying the tangent ratio, we can write the following equation involving the measure of ∠ X. tan X = YZ/XY ⇒ tan X = 5/4 Finally, applying the inverse tangent and using a calculator, we will find the required measure. m∠ X = arctan (5/4) ≈ 51.3^(∘)

Subchapter links

Exercises p.763-767