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Math Expert V
Joined: 02 Sep 2009
Posts: 58464
Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 60% (01:35) correct 40% (01:56) wrong based on 203 sessions

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Triangle XYZ is an isosceles right triangle. If side XY is longer than side YZ, and the area of the triangle is 16, what is the measure of side XY?

A. 4
B. 4√2
C. 8
D. 8√2
E. Cannot be determined from the information provided

Kudos for a correct solution.

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Math Expert V
Joined: 02 Aug 2009
Posts: 8023
Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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2
ans C ..8..
xy being larger means it is the hyp..
area =(1/2)*(yz)^2=16 or yz=4*\sqrt{2}..
therefore hyp=xy=8
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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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1/2 a^2 = 16
Side of triangle is 4 root 2
The ratio of sides of an isoceles triangle is 1:1:root 2.
There fore sides are in ratio of 4 root 2: 4 root 2: 8.
larger side is 8
Math Expert V
Joined: 02 Sep 2009
Posts: 58464
Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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Bunuel wrote:
Triangle XYZ is an isosceles right triangle. If side XY is longer than side YZ, and the area of the triangle is 16, what is the measure of side XY?

A. 4
B. 4√2
C. 8
D. 8√2
E. Cannot be determined from the information provided

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

While the stimulus doesn't seem to contain a ton of information, the terms "isosceles" and "right" that define the triangle make that one value (Area = 16) go a long way. Since you know that XY is longer than YZ, then XY must be the hypotenuse and sides YZ and XZ must be the shorter sides. Meaning, also that side XY will be equal to side YZ√2.

With an isosceles right triangle, you also know that the area (which here is 16) is equal to 1/2*(shorterside)^2. So that means that here 16=1/2*s^2 and that one of the shorter sides then measures 4√2. Since the hypotenuse, then is s√2, you'll multiply 4√2(√2) to find that side XY measures 8.
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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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Refer diagram below:

Attachment: tri.png [ 1.85 KiB | Viewed 6950 times ]

We require to find hypotenuse XY of the triangle

Area $$= \frac{1}{2} a^2 = 16$$

a $$= 4\sqrt{2}$$

XY $$= \sqrt{(4\sqrt{2})^2 + (4\sqrt{2})^2} = \sqrt{64} = 8$$
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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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Bunuel wrote:
Triangle XYZ is an isosceles right triangle. If side XY is longer than side YZ, and the area of the triangle is 16, what is the measure of side XY?

A. 4
B. 4√2
C. 8
D. 8√2
E. Cannot be determined from the information provided

Kudos for a correct solution.

In an isosceles right triangle, the two legs are equal in length and the hypotenuse is sqrt(2) times the length of the legs. If the side XY is longer than side YZ, XY must be the hypotenuse.

Area of isosceles right triangle $$= (1/2)*side^2 = 16$$
$$Side = 4\sqrt{2}$$

Hypotenuse $$XY = \sqrt{2}*4\sqrt{2} = 8$$

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Joined: 12 Sep 2015
Posts: 4018
Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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Top Contributor
Bunuel wrote:
Triangle XYZ is an isosceles right triangle. If side XY is longer than side YZ, and the area of the triangle is 16, what is the measure of side XY?

A. 4
B. 4√2
C. 8
D. 8√2
E. Cannot be determined from the information provided

Kudos for a correct solution.

Triangle XYZ is an isosceles right triangle.
Let's sketch an isosceles right triangle: Side XY is longer than side YZ
Since the hypotenuse is the longest side of a right triangle, side XY must be the hypotenuse. Add this to our diagram: This also means the last remaining vertex must be Z: The area of the triangle is 16. What is the measure of side XY?
Let j = the length of side ZY,
Since ZY = ZX, we can see that side ZX must also halve length j Area of triangle = (base)(height)/2
So, we can write: 16 = (j)(j)/2
Simplify: 16 = j²/2
Multiply both sides by 2 to get: 32 = j²
Solve: j = √32

What is the measure of side XY?
Our diagram now looks like this. Applying the Pythagorean Theorem, we can write: (√32)² + (√32)² = c²
Simplify: 32 + 32 = c²
Simplify: 64 = c²
Solve: c = 8

Cheers,
Brent

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Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than  [#permalink]

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According to the give condition , XY is hypotenuse

So we have ,

1\2 *XZ*YZ=16

XZ^2 = 32 ( XZ=YZ , Isosceles Triangle)

XZ = 4*2^1/2

So XY = 8 ( apply Pythagoras Theorem )
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Giving Kudos is the best way to encourage and appreciate people Re: Triangle XYZ is an isosceles right triangle. If side XY is longer than   [#permalink] 09 Feb 2019, 21:40
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