讲一下第五题---Here is the extracted content from the image: **General Information:** * **Title:** 初一数学学科作业 1.1.2 探索勾股定理练习题 (Junior High Year 1 Math Subject Homework 1.1.2 Exploring Pythagorean Theorem Practice Questions) * **Question Setter:** 杨海红 (Yang Haihong) * **Reviewer:** 党珍 (Dang Zhen) * **Time:** 2025.8.4 * **Parent Signature:** (Blank space) --- **Question 1:** * **Question Stem:** 1. 在△ABC中，若∠B+∠C=90°，则___ * **Chart/Diagram Description:** * **Type:** Geometric figure (Triangle). * **Main Elements:** A triangle labeled A, B, C. A right-angle symbol is shown at vertex A, indicating that ∠A = 90°. * **Options:** * A. BC=AB+AC * B. BC²=AB²+BC² * C. AB=AC+BC * D. BC²=AB²+AC --- **Question 2:** * **Question Stem:** 2. 如图，在Rt△ABC中，∠ACB=90°，AB=4. 分别以AC, BC为直径作半圆，面积分别记为S₁, S₂, 则S₁+S₂的值等于( ) * **Chart/Diagram Description:** * **Type:** Geometric figure (Right-angled triangle with semicircles). * **Main Elements:** A right-angled triangle ABC with the right angle at C. A semicircle is drawn on side AC with AC as its diameter. Another semicircle is drawn on side BC with BC as its diameter. The hypotenuse AB is labeled with length '4'. * **Options:** * A. 2π * B. 3π * C. 4π * D. 8π * **Other Relevant Text (Handwritten Calculation Hint):** π/8 (BC² + AC²) = π/8 AB² = 16 --- **Question 3:** * **Question Stem:** 3. 一直角三角形的一条直角边长是6，另一条直角边与斜边的和是18，则直角三角形的面积是( ) * **Options:** * A. 8 * B. 48 * C. 24 * D. 30 --- **Question 4:** * **Question Stem:** 4. 如图，一轮船以12海里/时的速度从港口A出发向东南方向航行，另一轮船以5海里/时的速度同时从港口A出发向东北方向航行，离开港口2小时后两船相距( ) * **Chart/Diagram Description:** * **Type:** Diagram showing directions and distances. * **Main Elements:** A central point labeled A (representing a port). Two lines originate from A. One line extends towards the southeast, labeled with length '24'. Another line extends towards the northeast, labeled with length '10'. A dashed line connects the endpoints of these two lines, forming a triangle with A. A right-angle symbol is depicted between the southeast and northeast directions, indicating they are perpendicular. * **Options:** * A. 13海里 * B. 16海里 * C. 20海里 * D. 26海里 * **Other Relevant Text (Handwritten Note):** √24 --- **Question 5:** * **Question Stem:** 5. “赵爽弦图”巧妙地利用面积关系证明了勾股定理，是我国古代数学的骄傲. 如图所示的“赵爽弦图”是由四个全等的直角三角形和一个小正方形拼成的一个大正方形. 如图，设直角三角形较长直角边长为a，较短直角边长为b. 若大正方形面积是9，小正方形面积是1，则ab的值是( ) * **Chart/Diagram Description:** * **Type:** Geometric figure (Zhao Shuang's Gouxian Diagram). * **Main Elements:** A large square. Inside this large square, there is a smaller square rotated. The space between the large and small squares is filled by four congruent right-angled triangles. * **Options:** * A. 4 * B. 6 * C. 8 * D. 10 --- **Question 6:** * **Question Stem:** 6. 如图，已知∠B=∠C=∠D=∠E=90°，且BC=DE=8，EF=2AB=2CD，AB=3，则A、F两点间的距离是( ) * **Chart/Diagram Description:** * **Type:** Geometric figure (Broken line / Polyline). * **Main Elements:** A series of connected line segments forming a path: A to B, B to C, C to D, D to E, E to F. Right-angle symbols are shown at vertices B, C, D, and E, indicating that AB ⊥ BC, BC ⊥ CD, CD ⊥ DE, and DE ⊥ EF. The segments are arranged such that AB is vertical (up), BC is horizontal (right), CD is vertical (down), DE is horizontal (right), EF is vertical (up). Lengths are indicated: AB has a length of 3, BC has a length of 8, and DE has a length of 8. * **Options:** * A. 16 * B. 20 * C. 25 * D. 24 --- **Question 7:** * **Question Stem:** 7. 《九章算术》是我国古代第一部数学专著，它的出现标志着中国古代数学形成了完整的体系. “折竹抵地”问题源自《九章算术》中：今有竹高一丈，未折抵地，去根六尺，问折者几何？意思是：一根竹子，原高一丈（一丈=10尺），一阵风将竹子折断，某竹梢恰好抵地，抵地处离竹子底部6尺远，则折断处离地面的高度是( ) * **Options:** * A. 5.3尺 * B. 6.8尺 * C. 4.7尺 * D. 3.2尺 --- **Question 8:** * **Question Stem:** 8. 如图，在△ABC中，AB=10，AC=13，AD⊥BC，垂足为D，M为AD上任一点，则MC²-MB²等于( ) * **Chart/Diagram Description:** * **Type:** Geometric figure (Triangle with altitude). * **Main Elements:** A triangle ABC. An altitude AD is drawn from vertex A to side BC, with D being the foot of the perpendicular on BC. A point M is located on the segment AD. Side AB is labeled with length '10'. Side AC is labeled with length '13'. * **Options:** * A. 23 * B. 46 * C. 65 * D. 69 * **Other Relevant Text (Handwritten Formula):** MC² - MB² = AC² - AB² --- **Question 9:** * **Question Stem:** 9. 如图，△ABC中，∠C=90°，AD平分∠BAC，AB=5，AC=3，则BD的长是( ) * **Chart/Diagram Description:** * **Type:** Geometric figure (Right-angled triangle with angle bisector). * **Main Elements:** A right-angled triangle ABC with the right angle at C. A line segment AD is drawn from vertex A to side BC, where D is on BC. An arc symbol indicates that AD bisects angle BAC. Side AB is labeled with length '5'. Side AC is labeled with length '3'. * **Options:** * A. 2.5 * (The rest of the options are not fully visible in the image.) --- **Question 10:** * **Question Stem:** 10. 李明想知道学校旗杆的高，他发现旗杆上的绳子垂直到地面还多1m，当他把绳子的 (The question is cut off.)