In right triangle XYZ, \angle Y = 90^\circ and \angle X = 45^\circ. The hypotenuse XZ has … — ACT

In right triangle $XYZ$ , $\angle Y = 90^\circ$ and $\angle X = 45^\circ$ . The hypotenuse $XZ$ has a length of $12\sqrt{2}$ units. Point $W$ lies on leg $XY$ such that $XW = 7$ units. What is the length, in units, of segment $ZW$ ?

$13$ Answer
B
$\sqrt{119}$
C
$17$
D
$\sqrt{193}$
E
$\sqrt{313}$

Answer

The length of segment

ZW

13

units.

Because

\triangle XYZ

is an isosceles right triangle (

45^\circ-45^\circ-90^\circ

), its leg lengths

XY

and

YZ

are equal to the hypotenuse divided by

\sqrt{2}

, giving

12

. Segment

YW

12 - 7 = 5

. In right triangle

\triangle ZYW

, the hypotenuse

ZW = \sqrt{12^2 + 5^2} = 13

Step-by-Step Solution

Determine the leg lengths of

\triangle XYZ

using special right triangle properties.

XY = YZ = 12

In a

45^\circ-45^\circ-90^\circ

right triangle, the hypotenuse is

\text{leg} \cdot \sqrt{2}

. Given

XZ = 12\sqrt{2}

, each leg length is

12

Calculate the length of segment

YW

YW = 5

Since point

W

lies on leg

XY

and

XW = 7

YW = XY - XW = 12 - 7 = 5

Apply the Pythagorean Theorem to right triangle

\triangle ZYW

to solve for hypotenuse

ZW

ZW = 13

ZW = \sqrt{YZ^2 + YW^2} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13

Key Concept

Properties of

45^\circ-45^\circ-90^\circ

special right triangles and multi-step applications of the Pythagorean Theorem.

Estimated Time:1m 0s

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