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In isosceles right A SPE, the measure of S= 90 and length SE = 12 V Rectangle PLCE with diagonal LE drawn forms a 30 angle at ZLEP Find the length of PL L

In isosceles right A SPE, the measure of S= 90 and length SE = 12 V Rectangle PLCE with diagonal LE drawn forms a 30 angle at ZLEP Find the length of PL L

Elementary Geometry For College Students, 7e
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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**Transcription for Educational Website** --- ### Understanding Right Triangles and Rectangles through Geometry In this lesson, we will explore the properties of an isosceles right triangle and a rectangle that share a common diagonal. #### Problem Statement: In isosceles right triangle \( \triangle SPE \), the measure of \( \angle S = 90^\circ \) and the length \( SE = 12\sqrt{2} \). Rectangle \( PLCE \) has a diagonal \( LE \) drawn that forms a \( 30^\circ \) angle at \( \angle LEP \). Find the length of \( \overline{PL} \). ![Geometric Figure](image link here) #### Diagram Explanation: The figure consists of an isosceles right triangle and a rectangle: - \( \triangle SPE \) with \( \angle S = 90^\circ \), \( S \) at the right angle. - The hypotenuse \( SE \) is given as \( 12\sqrt{2} \). - Rectangle \( PLCE \) is constructed such that its diagonal \( LE \) forms a \( 30^\circ \) angle with side \( PE \). Our objective is to find the length of \( \overline{PL} \). #### Steps to Solve: 1. **Understand the Relationship in the Isosceles Right Triangle:** - For \( \triangle SPE \), since it's isosceles and right-angled, the lengths of \( SP \) and \( EP \) are equal. - Given \( SE \) (the hypotenuse) is \( 12\sqrt{2} \), use the Pythagorean Theorem: \( SE^2 = SP^2 + EP^2 \). 2. **Calculate the Legs of the Triangle:** - Since \( SP = EP \) in an isosceles right triangle, let \( x \) be the length of each leg. - Thus, \( SE = x\sqrt{2} \). Given \( SE = 12\sqrt{2} \): \[ x\sqrt{2} = 12\sqrt{2} \implies x = 12 \] Hence, \( SP = EP = 12 \). 3. **Analyze Rectangle \( PLCE \):** - Given \( \angle LEP
Transcribed Image Text:**Transcription for Educational Website** --- ### Understanding Right Triangles and Rectangles through Geometry In this lesson, we will explore the properties of an isosceles right triangle and a rectangle that share a common diagonal. #### Problem Statement: In isosceles right triangle \( \triangle SPE \), the measure of \( \angle S = 90^\circ \) and the length \( SE = 12\sqrt{2} \). Rectangle \( PLCE \) has a diagonal \( LE \) drawn that forms a \( 30^\circ \) angle at \( \angle LEP \). Find the length of \( \overline{PL} \). ![Geometric Figure](image link here) #### Diagram Explanation: The figure consists of an isosceles right triangle and a rectangle: - \( \triangle SPE \) with \( \angle S = 90^\circ \), \( S \) at the right angle. - The hypotenuse \( SE \) is given as \( 12\sqrt{2} \). - Rectangle \( PLCE \) is constructed such that its diagonal \( LE \) forms a \( 30^\circ \) angle with side \( PE \). Our objective is to find the length of \( \overline{PL} \). #### Steps to Solve: 1. **Understand the Relationship in the Isosceles Right Triangle:** - For \( \triangle SPE \), since it's isosceles and right-angled, the lengths of \( SP \) and \( EP \) are equal. - Given \( SE \) (the hypotenuse) is \( 12\sqrt{2} \), use the Pythagorean Theorem: \( SE^2 = SP^2 + EP^2 \). 2. **Calculate the Legs of the Triangle:** - Since \( SP = EP \) in an isosceles right triangle, let \( x \) be the length of each leg. - Thus, \( SE = x\sqrt{2} \). Given \( SE = 12\sqrt{2} \): \[ x\sqrt{2} = 12\sqrt{2} \implies x = 12 \] Hence, \( SP = EP = 12 \). 3. **Analyze Rectangle \( PLCE \):** - Given \( \angle LEP
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