Question

1) The rectangle below has a perimeter of 38cm. Find x.

X
2x + 7
2) Find the largest angle
2x + 15
2x + 30
-x + 17
4x + 19
cm
(2 Marks)

Answer 1

The answers are attached to this response.

Explanation:

1.

⭐Properties of a rectangle

• the opposite sides are parallel and congruent (the same)
• perimeter = the sum of all the sides
• perimeter = 2(length)+2(width)

Using the properties of a rectangle, let's form an equation of the perimeter to solve for x.

perimeter = 2(length) + 2(width)

38 = 2(2x+7) + 2(x)

38 = 4x + 14 + 2x . . . . . . . . . distribute the coefficients of the expressions

38 = 6x + 14 . . . . . . . . . . . . . add like terms

24 = 6x . . . . . . . . . . . . . . . . . subtract 14 from LHS and RHS

∴ 4 = x . . . . . . . . . . . . . . . . . . divide LHS and RHS by 6

2.

The figure shown makes the shape of a kite.

⭐Properties of a kite

• the internal angles have a sum of 360°

Using the properties of a kite, let's form an equation of the sum of the internal angles to solve for x, and then substitute x into the largest angle expression to find its angle measure.

(∠1) + (∠2) + (∠3) + (∠4) = 360°

(2x + 15) + (2x+30) + (4x+19) + (x+17) = 360

9x + 81 = 360 . . . . . . . add like terms

9x = 279 . . . . . . . . . . . subtract 81 from LHS and RHS

x = 31 . . . . . . . . . . . . . . divide LHS and RHS by 9

Substitute x into the largest angle expression to find its angle measure.

4(31) + 19 = 143°

∴ 143° is the largest angle

Answer 2

1. 4 cm
2. 143°

Explanation:

You want the value of x that makes the perimeter of a rectangle be 38 cm when sides are x and (2x+7). And you want the largest angle in the quadrilateral with angles x+17, 2x+15, 2x+30, and 4x+19.

1. Perimeter

The perimeter of a rectangle is given by the formula ...

P = 2(L +W)

Substituting the given values, we have ...

38 = 2((2x+7) +x)

Dividing by 2 gives ...

19 = 3x +7

12 = 3x . . . . . . . subtract 7

4 = x . . . . . . . divide by 3

The value of x is 4 cm.

2. Angles

The sum of the angles of a quadrilateral is 360°. This tells us ...

360 = (x +17) +(2x +15) +(2x +30) +(4x +19)

360 = 9x +81 . . . . . . . collect terms

40 = x +9 . . . . . . divide by 9

31 = x . . . . . . . subtract 9

The largest angle is 4x+19, or 4(31) +19 = 124 +19 = 143.

The largest angle is 143°.