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ISEE Upper Level Math : How to find if right triangles are similar

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Example Question #1 : Right Triangles

$\Delta ABC \sim \Delta DEF$ $\displaystyle \Delta ABC \sim \Delta DEF$

$\angle B$ $\displaystyle \angle B$ is a right angle; DE = 21 $\displaystyle DE = 21$, DF = 42 $\displaystyle DF = 42$.

Which is the greater quantity?

(a) $m \angle A$ $\displaystyle m \angle A$

(b) $60 ^{\circ }$ $\displaystyle 60 ^{\circ }$

Possible Answers:

(b) is greater.

(a) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

$\Delta ABC \sim \Delta DEF$ $\displaystyle \Delta ABC \sim \Delta DEF$. Corresponding angles of similar triangles are congruent, so since $\angle B$ $\displaystyle \angle B$ is a right angle, so is $\angle E$ $\displaystyle \angle E$.

The hypotenuse $\overline{DF}$ $\displaystyle \overline{DF}$ of $\Delta DEF$ $\displaystyle \Delta DEF$ is twice as long as leg $\overline{DE}$ $\displaystyle \overline{DE}$; by the $30 ^{\circ } - 60^{\circ } -90^{\circ }$ $\displaystyle 30 ^{\circ } - 60^{\circ } -90^{\circ }$ Theorem, $m \angle D = 60 ^{\circ }$ $\displaystyle m \angle D = 60 ^{\circ }$. Again, by similiarity,

$m \angle A = m \angle D = 60 ^{\circ }$ $\displaystyle m \angle A = m \angle D = 60 ^{\circ }$.

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ISEE Upper Level Math : How to find if right triangles are similar

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Question #1 : Right Triangles

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