Triangle Area Calculation and Side Length
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3.
$$x = ?$$
(A shaded triangle $ABC$ with base $|BC| = 15$ and height $|AH| = 4$ is shown. In right triangle $ADC$, angle $D$ is $90^\circ$. $A$ is a point on $BD$. $|AB| = 10$, $|CD| = x$.)
A) 6
B) 8
C) 10
D) 12
E) 14
This question includes visual content: A geometric figure labeled with points A, B, C, D, and H. There is a triangle ABC where a perpendicular segment AH is drawn from vertex A to the line segment BC. The length of AH is given as 4. The length of BC is given as 15. The length of segment AB is labeled as 10. There is an external vertex D such that angle ADC is 90 degrees (indicated by a square symbol). Vertex A lies on the line segment BD. The length of segment CD is labeled as x. The triangle ABC is shaded gray. The question mark asks $x = ?$.
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Step by Step Written Solution
Let's find the value of x in this geometry problem. We have a shaded triangle ABC and an external right triangle ADC.
Area of Triangle Calculations
Observe triangle ABC. We are given its base BC, which is fifteen units long, and its altitude AH, which is four units long.
The area of triangle ABC can be calculated using the formula: one half times base times height.
Substituting fifteen for the base and four for the height, we get thirty square units.
Now, let's look at the same triangle ABC from a different perspective. If we take side AB as the base, the length of the altitude drawn to this base is actually the segment CD.
Notice that CD is perpendicular to the extension of AB at point D. This makes CD, which is labeled as x, the altitude for base AB, which is ten.
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