📐 geometry

1. **Stating the problem:** We are given angles $x^\circ$, $w^\circ$, $y^\circ$ with the condition $BA = BD$ and the ratio $x : y = 2 : 1$. We need to find the relationship between

1. The problem asks for the surface area of a right rectangular prism with dimensions 3 cm (height), 10 cm (width), and 11 cm (depth). 2. The formula for the surface area $A$ of a

1. **Problem Statement:** Find the surface area of the first cuboid with dimensions 4 cm by 4 cm by 7 cm. 2. **Formula:** The surface area $A$ of a cuboid with length $l$, width $w

1. **Problem:** Find the surface area of the first cuboid with length $8$ cm, width $5$ cm, and height $5$ cm. 2. **Formula:** The surface area $A$ of a cuboid is given by:

1. **Problem statement:** We have two circles with radii $a$ and $b$, and the distance between their centers is $c$. We want to find the area of their overlapping region. 2. **Form

1. **Problem statement:** We have two circles with radii $a$ and $b$ where $a > b$, and the distance between their centers is $c$. We want to find the area of their overlap. 2. **K

1. **Problem statement:** Find the percentage of the 7 by 7 grid that is shaded. 2. **Total squares:** The grid has $7 \times 7 = 49$ squares.

1. **State the problem:** We have a right-angled triangle XYZ with right angle at Y. 2. **Given:**

1. **Problem statement:** We have a right-angled triangle PQR with the right angle at R. The hypotenuse PQ is 50 cm, and one leg QR is 48 cm. We need to find the length of the othe

1. The problem involves understanding why angle CFA is not possible in step 4. 2. To analyze this, we need to consider the geometric constraints and definitions of the points C, F,

1. **Problem statement:** We have a circle with center $O$. Lines $BC$ and $DE$ are tangents to the circle at points $B$ and $D$ respectively. We know $\angle AOB = 52^\circ$ and $

1. **Problem statement:** Show that triangles ABC and FEC are similar and explain why line OB is parallel to line PE.

1. The problem is to find the area, but the shape or dimensions are not specified. 2. To find the area, we need to know the type of figure (e.g., rectangle, triangle, circle) and i

1. **Stating the problem:** We have a trapezoid with height $h=5$ cm, bottom base $a=5$ cm, and one slant side $c=3$ cm. There is a right angle at the intersection of height and ba

1. **State the problem:** We have three points A, B, and C on a grid. The path from A to B is vertical, and the path from B to C is diagonal. The cost per metre of the path is fixe

1. **Problem statement:** We need to find the length of side $QR$ in right triangle $PQR$ where the right angle is at vertex $P$. Given sides are $PQ = 8$ cm and $PR = 3$ cm. 2. **

1. **State the problem:** We need to find the length of side $YZ$ in a right triangle $XYZ$ where $XY = 5$ cm and $XZ = 19$ cm. The right angle is at $Y$, so $YZ$ is perpendicular

1. The problem asks to reflect the shape ABCD across the x-axis and write the transformation rule. 2. Reflection across the x-axis changes the y-coordinate to its opposite while ke

1. **Match the quotes to the correct reflection rules:** - D. Reflection across the x-axis: $(x, y) \to (x, -y)$ matches quote D: "To reflect the figure, all x values will remain t

1. **State the problem:** A barrel shaped like a cylinder with diameter 3.1 ft is rolled up a ramp 379 ft long. We need to find how many times the barrel turns. 2. **Formula used:*

1. **Problem:** Determine the area of a circle with radius 6 cm using the formula $A = \pi r^2$. 2. **Formula:** The area of a circle is given by