8) Now he posted this question and asked to find the altitude of the triangle CD. What should be the length of CD? In the diagram below of right triangle ABC, altitude CD is drawn to hypotenuse AB. A 3 D 12
8) Now he posted this question and asked to find the altitude of the triangle CD. What should be the length of CD? In the diagram below of right triangle ABC, altitude CD is drawn to hypotenuse AB. A 3 D 12
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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![### Finding the Altitude of Triangle CD
**Problem Statement:**
Now he posted this question and asked to find the altitude of the triangle CD. What should be the length of CD?
**Diagram Description:**
In the diagram below of right triangle \( \triangle ABC \), altitude \( CD \) is drawn to hypotenuse \( AB \).
\[
\begin{tikzpicture}
\draw[thick] (0,0) node[left] {A} -- (5,0) node[right] {B} -- (1.5,3) node[above] {C} -- cycle;
\draw (1.5,0) -- (1.5,1.5);
\draw (1.5,3) -- (1.5,0) node[left] {D};
\node at (0.75,0) {3};
\node at (3.5,0) {12};
\end{tikzpicture}
\]
\( CD \) forms a right angle with \( AB \).
**Mathematical Explanation:**
To find the length of \( CD \), we can use the properties of right triangles and similar triangles.
Given:
- \( AD = 3 \)
- \( DB = 12 \)
We know that \( AB = AD + DB = 3 + 12 = 15 \).
Using the altitude-on-hypotenuse theorem, which states that the altitude to the hypotenuse of a right triangle is the geometric mean of the segments it divides the hypotenuse into:
\[
CD^2 = AD \cdot DB
\]
\[
CD^2 = 3 \cdot 12
\]
\[
CD^2 = 36
\]
\[
CD = \sqrt{36}
\]
\[
CD = 6
\]
Hence, the length of \( CD \) is 6 units.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff8c464c8-9816-4ef4-8009-bf3331731bd2%2F0a1e6634-1570-43ef-87ca-53edbd834dc1%2Fsdidrfw_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Altitude of Triangle CD
**Problem Statement:**
Now he posted this question and asked to find the altitude of the triangle CD. What should be the length of CD?
**Diagram Description:**
In the diagram below of right triangle \( \triangle ABC \), altitude \( CD \) is drawn to hypotenuse \( AB \).
\[
\begin{tikzpicture}
\draw[thick] (0,0) node[left] {A} -- (5,0) node[right] {B} -- (1.5,3) node[above] {C} -- cycle;
\draw (1.5,0) -- (1.5,1.5);
\draw (1.5,3) -- (1.5,0) node[left] {D};
\node at (0.75,0) {3};
\node at (3.5,0) {12};
\end{tikzpicture}
\]
\( CD \) forms a right angle with \( AB \).
**Mathematical Explanation:**
To find the length of \( CD \), we can use the properties of right triangles and similar triangles.
Given:
- \( AD = 3 \)
- \( DB = 12 \)
We know that \( AB = AD + DB = 3 + 12 = 15 \).
Using the altitude-on-hypotenuse theorem, which states that the altitude to the hypotenuse of a right triangle is the geometric mean of the segments it divides the hypotenuse into:
\[
CD^2 = AD \cdot DB
\]
\[
CD^2 = 3 \cdot 12
\]
\[
CD^2 = 36
\]
\[
CD = \sqrt{36}
\]
\[
CD = 6
\]
Hence, the length of \( CD \) is 6 units.
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