https://file.aiquickdraw.com/mathbot/file/1728614612302lhzim38s.jpg

Let's solve the three sub-questions one by one from the image:

### i) If \( x = 1 \) and \( y = 1 \), and the coordinates of \( A \) is \( x \) and \( B \) is \( y \), find \( d(A,B) \).

Given that the coordinate of point \( A \) is \( (1, 0) \) and the coordinate of point \( B \) is \( (0, 1) \) (since the coordinates are defined by \( x \) and \( y \), where \( x = 1 \) and \( y = 1 \)).

The distance formula is:
\[
d(A, B) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substituting the coordinates \( A(1, 0) \) and \( B(0, 1) \), we get:
\[
d(A, B) = \sqrt{(0 - 1)^2 + (1 - 0)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \approx 1.414
\]

### ii) In \( \triangle PQR \), \( \angle P = 70^\circ \) and \( \angle Q = 65^\circ \). Find \( \angle R \).

The sum of the interior angles of a triangle is always \( 180^\circ \). Therefore:
\[
\angle P + \angle Q + \angle R = 180^\circ
\]
Substituting the values of \( \angle P \) and \( \angle Q \):
\[
70^\circ + 65^\circ + \angle R = 180^\circ
\]
\[
135^\circ + \angle R = 180^\circ
\]
\[
\angle R = 180^\circ - 135^\circ = 45^\circ
\]

### iii) Two parallel lines are intersected by a transversal. If the measure of one of the alternate interior angles is \( 75^\circ \), find the measure of the other angle.

Since the lines are parallel and alternate interior angles are always congruent (equal), the measure of the other angle is also \( 75^\circ \).

---

Would you like further details on any of these solutions?

Here are 5 related questions for further practice:
1. What is the general form of the distance formula for two points?
2. How do you calculate the third angle in a triangle if two angles are known?
3. What is the sum of interior angles in any polygon, and how does it change with the number of sides?
4. How can you prove that alternate interior angles are congruent using properties of parallel lines?
5. What is the difference between corresponding angles and alternate interior angles?

**Tip:** Always double-check angle measures when solving geometric problems to ensure they sum up correctly, especially in triangles or when using the properties of parallel lines.

i) If x = 1 and y = 1 and the coordinate of A is x and B is y, find d(A,B). ii) In ∆PQR, ∠P = 70° and ∠Q = 65°, find ∠R. iii) Two parallel lines are intersected by a transversal. If one alternate interior angle is 75°, find the measure of the other angle.

Solve a geometric problem involving the distance formula, finding the third angle of a triangle, and alternate interior angles of parallel lines. Includes step-by-step solutions for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation