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# If the hypotenuse of isosceles right triangle ABC has the same length

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If the hypotenuse of isosceles right triangle ABC has the same length [#permalink]

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02 Feb 2017, 23:33
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If the hypotenuse of isosceles right triangle ABC has the same length as the height of equilateral triangle DEF, what is the ratio of a leg of triangle ABC to a side of triangle DEF?

A. $$\sqrt{2} / 2$$

B. $$\sqrt{3} / 2$$

C. $$\sqrt{3} /[ 2 *\sqrt{2}]$$

D. $$\sqrt{2} / \sqrt{3}$$

E. $$\frac{3}{2}$$
[Reveal] Spoiler: OA

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Last edited by hazelnut on 03 Feb 2017, 04:49, edited 14 times in total.
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Re: If the hypotenuse of isosceles right triangle ABC has the same length [#permalink]

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03 Feb 2017, 00:18
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Right angle isosceles triangle- Sides are in ration 1:1:root 2
Given hyp. of isosceles is the height of equilateral- i.e height of equilateral is root 2.
Half of equilateral triangle is 30-60-90 triangle whose sides are in ratio- 1:root 3: 2
Leg of ABC : Side of DEF = 1 : (2*root 2/root 3) = root 3 : 2 (root 2)
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Re: If the hypotenuse of isosceles right triangle ABC has the same length [#permalink]

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03 Feb 2018, 14:09
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hazelnut wrote:
If the hypotenuse of isosceles right triangle ABC has the same length as the height of equilateral triangle DEF, what is the ratio of a leg of triangle ABC to a side of triangle DEF?

A. $$\sqrt{2} / 2$$

B. $$\sqrt{3} / 2$$

C. $$\sqrt{3} /[ 2 *\sqrt{2}]$$

D. $$\sqrt{2} / \sqrt{3}$$

E. $$\frac{3}{2}$$

Attachment:

tris.png [ 24.18 KiB | Viewed 303 times ]

Isosceles right triangle
An isosceles right triangle has angle measures 45-45-90
Side lengths opposite those angles are in ratio
$$x: x: x\sqrt{2}$$ or
$$1: 1: \sqrt{2}$$

Equilateral triangle
Any altitude of an equilateral triangle creates two congruent triangles with angle measures 30-60-90
Side lengths opposite those angles are in ratio
$$x: x\sqrt{3}: 2x$$ or
$$1: \sqrt{3}: 2$$

The hypotenuse of isosceles right ∆ ABC = the height (altitude) of equilateral ∆ DEF
What is the ratio of a leg of ∆ ABC to a side of ∆ DEF?

1) Assign a value for a leg of ∆ ABC. Find hypotenuse BC
Let a leg of the isosceles right triangle, opposite a 45° angle $$= 1 = x$$
Hypotenuse BC, opposite the 90° angle = $$x\sqrt{2} = (1*\sqrt{2}) = \sqrt{2}$$
BC = $$\sqrt{2}$$

2) Find side length of ∆ DEF from BC and 30-60-90 ∆ side ratios
BC = EG
Height EG, opposite the 60° angle = $$x\sqrt{3}$$
Height EG also = $$\sqrt{2}$$

$$x\sqrt{3} = \sqrt{2}$$

$$x = \frac{\sqrt{2}}{\sqrt{3}}$$

Side of ∆ DEF = $$(2x) = (2 *\frac{\sqrt{2}}{\sqrt{3}})= \frac{2\sqrt{2}}{\sqrt{3}}$$

3) Ratio of a leg of ∆ ABC to a side of ∆ DEF?

Leg of ∆ ABC = 1 (from above)

Side of ∆ DEF = $$\frac{2\sqrt{2}}{\sqrt{3}}$$

$$\frac{1}{(\frac{2√2}{√3})} = \frac{√3}{2√2}$$

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Re: If the hypotenuse of isosceles right triangle ABC has the same length   [#permalink] 03 Feb 2018, 14:09
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