Finding Unknowns in Triangles using Similarity and Euclidean Theorems
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Find the unknown numbers $x$, $y$ and $z$ of the following figures.
a. A right triangle with hypotenuse segments $9$ and $5$. Altitude is $z$. Legs are $x$ and $y$. Total hypotenuse length indicated is $15$.
b. A triangle with a transversal parallel to the base. Top side segment is $10$, bottom side segment is $x$. Base is $12$ and parallel segment is $8$.
c. A triangle with a transversal parallel to the base. Side segments are $12$ and $8$ on the left. Total side length $14$ on the right with a segment $x$.
d. A right triangle with altitude $8$ dividing the hypotenuse into segments $4$ and $x$. Legs are $y$ and $z$.
This question includes visual content: The image shows four separate geometric diagrams labeled a, b, c, and d. Diagram a: A right-angled triangle with an altitude dropped from the right angle to the hypotenuse. The hypotenuse is divided into two segments of length $9$ and $5$ (total length $14$ is shown below it as $15$, which might be a typo in the book or intended measurement). The altitude is $z$, and the two other sides are $y$ and $x$. Diagram b: A triangle with a line segment parallel to the base inside it. The sides of the larger triangle are $10$ and $x$. The parallel segments have lengths $8$ and $12$. Diagram c: A triangle with a line segment parallel to the base. The left side is divided into segments of $12$ and $8$. The right side has a total length of $14$ and a segment marked $x$. Diagram d: A right-angled triangle with an altitude drawn to the hypotenuse. The segments of the hypotenuse are $4$ and $x$. The altitude is $8$. The other legs are $y$ and $z$.
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Step by Step Written Solution
Let's find the unknown values x, y, and z from these geometric figures. We will focus on the right-angled relationships and similar triangles shown.
Geometry: Solving for Unknowns
Starting with figure a, we have a large right-angled triangle with an altitude drawn to the hypotenuse. The hypotenuse length is fifteen, and it is split into segments of five and ten.
Figure a Analysis
First, we can find z using the Geometric Mean Theorem for the altitude. The square of the altitude z equals the product of the two segments of the hypotenuse.
So, z squared is fifty. Taking the square root, z equals five times the square root of two.
Next, let's find x using the right triangle on the right side. By the Pythagorean theorem, x squared equals z squared plus five squared.
Substituting fifty for z squared, we get x squared equals seventy-five. Thus, x equals five times the square root of three.
Finally, we find y using the left-hand triangle. y squared equals z squared plus ten squared.
This gives us fifty plus one hundred, which is one hundred and fifty. So, y equals five times the square root of six.
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