Question

1. in right triangle abc, altitude cd is drawn to its hypotenuse.

select all triangles which must be similar to triangle abc.
a. abc
b. acd
c. bcd
d. bdc
e. cad
f. cbd

1. in right triangle abc, altitude cd with length h is drawn to its hypotenuse. we also know ad = 12 and db = 3.

what is the value of h?

1. in triangle abc (not a right triangle), altitude cd is drawn to side ab. the length of ab is c.

which of the following statements must be true?
a. the measure of angle acb is the same measure as angle b.
b. ( b^2 = c^2 + a^2 ).
c. triangle adc is similar to triangle acb.
d. the area of triangle abc equals ( \frac{1}{2}h cdot c ).

Explanation:

Problem 1

Step1: Recall Similar Triangles in Right Triangles

In a right triangle, when an altitude is drawn to the hypotenuse, the two smaller triangles formed are similar to the original triangle and to each other. So, $\triangle ABC \sim \triangle ACD \sim \triangle BCD$ (or $\triangle CBD$ as $BCD$ and $CBD$ are the same triangle, just named differently). Also, a triangle is similar to itself, so $\triangle ABC$ is similar to $\triangle ABC$.

Step2: Analyze Each Option

• Option A: $\triangle ABC$ is similar to itself (reflexive property of similarity), so this is valid.
• Option B: $\triangle ACD$ is one of the smaller triangles formed by the altitude, so it is similar to $\triangle ABC$.
• Option C: $\triangle BCD$ (same as $\triangle CBD$) is the other smaller triangle, so it is similar to $\triangle ABC$.
• Option D: $\triangle BDC$ is the same as $\triangle BCD$, so it is similar to $\triangle ABC$.
• Option E: $\triangle CAD$ is the same as $\triangle ACD$, so it is similar to $\triangle ABC$.
• Option F: $\triangle CBD$ is the same as $\triangle BCD$, so it is similar to $\triangle ABC$.

Step1: Recall the Geometric Mean Theorem (Altitude-on-Hypotenuse Theorem)

In a right triangle, the length of the altitude drawn to the hypotenuse is the geometric mean of the lengths of the two segments it divides the hypotenuse into. So, $h^2 = AD \times DB$.

Step2: Substitute the Given Values

We know that $AD = 12$ and $DB = 3$. Substituting these values into the formula, we get $h^2 = 12 \times 3$.

Step3: Calculate $h$

First, calculate $12 \times 3 = 36$. Then, take the square root of both sides: $h = \sqrt{36} = 6$.

Step1: Analyze Each Option

• Option A: There is no information given to suggest that $\angle ACB$ (which is $75^\circ$) is equal to $\angle B$. So this is not necessarily true.
• Option B: The Pythagorean theorem ($b^2 = c^2 + a^2$) holds only for right triangles. Since $\triangle ABC$ is not a right triangle, this is false.
• Option C: For two triangles to be similar, their corresponding angles must be equal. $\triangle ADC$ is a right triangle (since $CD$ is an altitude), but $\triangle ACB$ is not a right triangle, so they cannot be similar.
• Option D: The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$. Here, the base is $AB = c$ and the height is $CD = h$, so the area of $\triangle ABC$ is $\frac{1}{2} \times c \times h$, which is $\frac{1}{2}h \cdot c$. This is true.

Answer:

A, B, C, D, E, F

Problem 2