Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN: 9781337614085
Author: Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher: Cengage,
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Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made p…
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A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
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Find the value of x

### Solving for Variables in Right Triangles #### Triangle Diagram Explanation The given diagram is composed of two right-angled triangles. Here's the detailed description and analysis: 1. **Larger Triangle:** - One of the legs is 21 units long. - The hypotenuse is labeled as "x." - The height of the larger triangle connects perpendicularly to the hypotenuse of the smaller right triangle, forming the complete larger triangle. 2. **Smaller Triangle:** - One of the legs (adjacent to the right angle) is labeled "9." - The other leg (opposite to the right angle) is labeled "y." - The hypotenuse of the smaller right triangle is the same as one of the legs of the larger triangle. 3. **Right Angles:** - Both triangles have a right angle (90 degrees). To solve for the unknown variables \(x\) and \(y\), break the problem down using the Pythagorean theorem: **Steps:** 1. **Pythagorean Theorem Application to Larger Triangle:** \[ 21^2 + (y + 9)^2 = x^2 \] 2. **Pythagorean Theorem Application to Smaller Triangle:** \[ 9^2 + y^2 = x^2 \] Combine and solve these equations algebraically to find the values of \(x\) and \(y\). **Given Equation:** \[ x = 270 \] Using appropriate methods in geometry and algebra, such as substitution or simultaneous equations, these relationships can be solved to find precise lengths for \(y\). This approach helps in practicing and understanding basic trigonometric concepts applicable to right-angled triangles.
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Transcribed Image Text:### Solving for Variables in Right Triangles #### Triangle Diagram Explanation The given diagram is composed of two right-angled triangles. Here's the detailed description and analysis: 1. **Larger Triangle:** - One of the legs is 21 units long. - The hypotenuse is labeled as "x." - The height of the larger triangle connects perpendicularly to the hypotenuse of the smaller right triangle, forming the complete larger triangle. 2. **Smaller Triangle:** - One of the legs (adjacent to the right angle) is labeled "9." - The other leg (opposite to the right angle) is labeled "y." - The hypotenuse of the smaller right triangle is the same as one of the legs of the larger triangle. 3. **Right Angles:** - Both triangles have a right angle (90 degrees). To solve for the unknown variables \(x\) and \(y\), break the problem down using the Pythagorean theorem: **Steps:** 1. **Pythagorean Theorem Application to Larger Triangle:** \[ 21^2 + (y + 9)^2 = x^2 \] 2. **Pythagorean Theorem Application to Smaller Triangle:** \[ 9^2 + y^2 = x^2 \] Combine and solve these equations algebraically to find the values of \(x\) and \(y\). **Given Equation:** \[ x = 270 \] Using appropriate methods in geometry and algebra, such as substitution or simultaneous equations, these relationships can be solved to find precise lengths for \(y\). This approach helps in practicing and understanding basic trigonometric concepts applicable to right-angled triangles.
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