Let’s solve the problems step-by-step.

Question 25:

a. Draw X, Y axes and mark the points:

The points given are:

• (-3, 0), (3, 0), (6, 4), (0, 4)

These points can be plotted on the Cartesian coordinate system. The quadrilateral formed is:

1. Vertices: (-3, 0), (3, 0), (6, 4), (0, 4)
2. Shape: A trapezium, as one pair of opposite sides is parallel.

b. Find the area of the quadrilateral:

The vertices are:

• $A(-3, 0)$, $B(3, 0)$, $C(6, 4)$, $D(0, 4)$.

Use the Shoelace formula to find the area: $\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$

Substitute the coordinates: $\text{Area} = \frac{1}{2} \left| (-3)(0) + (3)(4) + (6)(4) + (0)(0) - \left( 0(3) + 0(6) + 4(0) + 4(-3) \right) \right|$ $\text{Area} = \frac{1}{2} \left| 0 + 12 + 24 + 0 - (0 + 0 + 0 - 12) \right|$ $\text{Area} = \frac{1}{2} \left| 36 + 12 \right| = \frac{1}{2} \cdot 48 = 24 \, \text{square units.}$

Question 26:

In the figure, PQ, QR, and PR are tangents to the circle.

• $\angle RAC = 65^\circ$, $\angle PAB = 55^\circ$

a. Find $\angle BAC$:

• In $\triangle ABC$, the sum of angles is $180^\circ$. Hence: $\angle BAC = 180^\circ - (\angle RAC + \angle PAB)$ $\angle BAC = 180^\circ - (65^\circ + 55^\circ) = 60^\circ.$

b. Find $\angle ABP$:

• $\angle ABP$ is vertically opposite to $\angle PAB$, so: $\angle ABP = 55^\circ.$

c. Find all the angles of $\triangle PQR$:

• Since PQ, QR, and PR are tangents, $\triangle PQR$ is formed by the external angles of the tangents.

The angles of $\triangle PQR$ can be calculated using the given circle-tangent geometry rules:

1. $\angle P = 180^\circ - \angle PAB = 180^\circ - 55^\circ = 125^\circ$
2. Similarly, find other angles by using symmetry or tangent angle properties.

Would you like detailed steps for part 26c? Let me know! 😊

Follow-Up Questions:

1. Can you calculate areas for shapes with different sets of vertices using the Shoelace formula?
2. Why does a quadrilateral with two parallel sides form a trapezium?
3. How are tangent properties useful in solving problems with circles and triangles?
4. What is the significance of external angles in geometry?
5. Can the Shoelace formula be used for polygons with more than four sides?

Tip:

When working with geometry problems, always check if properties like symmetry, parallelism, or tangency can simplify your calculations!