📐 geometry

1. Bài 19.1 a) Cho hình chóp S.ABCD, M, N lần lượt là trung điểm của AB và BC. Tìm giao tuyến của hai mặt phẳng (SAC) và (SBD).

1. **Problem Statement:** Given a circle with center $O$ passing through points $A$, $B$, and $C$, and a tangent line $PQ$ touching the circle at $A$. It is given that $\angle PAB

1. **Problem statement:** Given triangle ABC with points P, Q, R, and S such that QS is parallel to BA, QR is parallel to CA, and PQ = 10 cm. We need to find the product $PB \times

1. **Problem statement:** We have a right triangle with an altitude $d$ drawn to the hypotenuse $c$, dividing it into segments $p$ and $q$. The sides are labeled $a$, $b$, and the

1. **State the problem:** We need to find the area of the shaded portion formed by two semicircles with diameters 42 cm and 21 cm. 2. **Formula for the area of a semicircle:** The

1. **Problem Statement:** Find the total area of the shaded portions of two semicircles, each with diameter 42 cm. 2. **Formula for the area of a semicircle:**

1. **Problem Statement:** Given isosceles triangle $PQR$ with $PQ = PR$, points $S$ and $T$ lie on $PQ$ and $PR$ respectively such that $ST \parallel QR$. Perpendiculars from $S$ a

1. 题目描述：图中有一个矩形和两个半径分别为5和2的直角扇形。求两个阴影部分的面积之差。（π取3） 2. 公式和规则：

1. Bài toán: Cho đường tròn $(O)$ có đường kính $AB$, $I$ là trung điểm của $OA$. Kẻ dây $CD$ vuông góc với $AB$ tại $I$. Lấy điểm $M$ bất kỳ thuộc cung nhỏ $BC$. Gọi $H$ là giao đ

1) a) Énoncé : Trouver la mesure principale de $ (\overrightarrow{AB}, \overrightarrow{AC}) \equiv -\frac{15\pi}{4} \ [2\pi]$. Rappel : La mesure principale d'un angle est l'angle

1. The problem asks to find the total number of small unit cubes in the 3D block structure described. 2. Let's analyze the description step-by-step:

1. **Problem:** Find the total area of the L-shaped figure by dividing it into two rectangles. 2. **Formula:** Area of a rectangle = length \times width.

1. **Problem statement:** We have a right triangle with two equal sides labeled $a$ and two semicircles drawn on these sides. We want to find the sum of the diameters of both semic

1. **Stating the problem:** We have two parallel lines $d_1$ and $d_2$ with points $A, B, C, D, E$ arranged such that angles $m(ABC) = 150^\circ$, $m(BCD) = 100^\circ$, and we want

1. The problem states that there is a rectangular grid with 8 columns and 4 rows, making a total of $8 \times 4$ small squares. 2. The total number of small squares is calculated b

1. Let's clarify the context: typically, when you see a change from $r^2$ to $r^3$, it involves a formula where volume or a three-dimensional measure is considered instead of area

1. **Problem:** In triangle ABC, given that AD = DC = BC and \(\angle BCE = 96^\circ\), find \(\angle DCB\). 2. **Understanding the problem:**

1. **State the problem:** In triangle $PQR$, given $\angle Q = 50^\circ$ and $\angle R = 40^\circ$, $PS$ bisects $\angle QPR$, and $PT \perp QR$. Find the value of $\angle TPS + 2\

1. **Problem statement:** A ladder leans against a wall, touching the top of a fence 1.5 m high and 1 m away from the wall. We need to find the minimum length of the ladder. 2. **S

1. **State the problem:** Calculate the value of $c$ using the formula $$c^2 = 59^2 + 317^2 - 2(59)(317)\cos 83^\circ$$ 2. **Formula used:** This is the Law of Cosines formula, whi

1. **State the problem:** We have a triangle with three angles: 130°, (x + 5)°, and (x + 15)°. We need to find the value of $x$. 2. **Recall the triangle angle sum rule:** The sum