in exercises 3–6, show that the quadrilateral with the given vertices is a trapezoid. then decide whether it is isosceles. (see example 1.)
3. w(1, 4), x(1, 8), y(-3, 9), z(-3, 3)
4. d(-3, 3), e(-1, 1), f(1, -4), g(-3, 0)
5. m(-2, 0), n(0, 4), p(5, 4), q(8, 0)
6. h(1, 9), j(4, 2), k(5, 2), l(8, 9)

Question

Accepted Answer

Let's solve problem 3: $ W(1, 4) $, $ X(1, 8) $, $ Y(-3, 9) $, $ Z(-3, 3) $

### Step 1: Recall the definition of a trapezoid and isosceles trapezoid
A trapezoid is a quadrilateral with at least one pair of parallel sides. The slope formula between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ m=\frac{y_2 - y_1}{x_2 - x_1} $. If two sides have the same slope, they are parallel. An isosceles trapezoid has the non - parallel sides (legs) congruent. The distance formula between two points $ (x_1, y_1) $ and $ (x_2, y_2) $ is $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $

### Step 2: Calculate the slopes of the sides
- Slope of $ WX $: For $ W(1,4) $ and $ X(1,8) $, using the slope formula $ m=\frac{y_2 - y_1}{x_2 - x_1} $, we have $ m_{WX}=\frac{8 - 4}{1 - 1}=\frac{4}{0} $, which is undefined. This means $ WX $ is a vertical line.
- Slope of $ XY $: For $ X(1,8) $ and $ Y(-3,9) $, $ m_{XY}=\frac{9 - 8}{-3 - 1}=\frac{1}{-4}=-\frac{1}{4} $
- Slope of $ YZ $: For $ Y(-3,9) $ and $ Z(-3,3) $, $ m_{YZ}=\frac{3 - 9}{-3+3}=\frac{-6}{0} $, which is undefined. This means $ YZ $ is a vertical line.
- Slope of $ ZW $: For $ Z(-3,3) $ and $ W(1,4) $, $ m_{ZW}=\frac{4 - 3}{1 + 3}=\frac{1}{4} $

Since $ m_{WX} $ (undefined, vertical line) and $ m_{YZ} $ (undefined, vertical line) are equal (both vertical), $ WX\parallel YZ $. So, the quadrilateral $ WXYZ $ is a trapezoid.

### Step 3: Check if it is isosceles (check the lengths of the legs $ XY $ and $ ZW $)
- Length of $ XY $: Using the distance formula $ d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2} $, for $ X(1,8) $ and $ Y(-3,9) $, $ d_{XY}=\sqrt{(-3 - 1)^2+(9 - 8)^2}=\sqrt{(-4)^2+1^2}=\sqrt{16 + 1}=\sqrt{17} $
- Length of $ ZW $: For $ Z(-3,3) $ and $ W(1,4) $, $ d_{ZW}=\sqrt{(1 + 3)^2+(4 - 3)^2}=\sqrt{4^2+1^2}=\sqrt{16 + 1}=\sqrt{17} $

Since $ d_{XY}=d_{ZW}=\sqrt{17} $, the legs are congruent. So, the trapezoid is isosceles.

### Final Answer for problem 3:
The quadrilateral with vertices $ W(1,4) $, $ X(1,8) $, $ Y(-3,9) $, $ Z(-3,3) $ is a trapezoid (because $ WX\parallel YZ $) and it is isosceles (because the legs $ XY $ and $ ZW $ are congruent).

Let's solve problem 4: $ D(-3, 3) $, $ E(-1, 1) $, $ F(1, -4) $, $ G(-3, 0) $

### Step 1: Calculate the slopes of the sides
- Slope of $ DE $: For $ D(-3,3) $ and $ E(-1,1) $, $ m_{DE}=\frac{1 - 3}{-1+3}=\frac{-2}{2}=-1 $
- Slope of $ EF $: For $ E(-1,1) $ and $ F(1,-4) $, $ m_{EF}=\frac{-4 - 1}{1 + 1}=\frac{-5}{2}=-\frac{5}{2} $
- Slope of $ FG $: For $ F(1,-4) $ and $ G(-3,0) $, $ m_{FG}=\frac{0 + 4}{-3 - 1}=\frac{4}{-4}=-1 $
- Slope of $ GD $: For $ G(-3,0) $ and $ D(-3,3) $, $ m_{GD}=\frac{3 - 0}{-3+3}=\frac{3}{0} $, which is undefined.

Since $ m_{DE}=m_{FG}=-1 $, $ DE\parallel FG $. So, the quadrilateral $ DEFG $ is a trapezoid.

### Step 2: Check if it is isosceles (check the lengths of the legs $ EF $ and $ GD $)
- Length of $ EF $: For $ E(-1,1) $ and $ F(1,-4) $, $ d_{EF}=\sqrt{(1 + 1)^2+(-4 - 1)^2}=\sqrt{2^2+(-5)^2}=\sqrt{4 + 25}=\sqrt{29} $
- Length of $ GD $: For $ G(-3,0) $ and $ D(-3,3) $, $ d_{GD}=\sqrt{(-3 + 3)^2+(3 - 0)^2}=\sqrt{0 + 9}=3 $

Since $ \sqrt{29}
eq3 $, the legs are not congruent. So, the trapezoid is not isosceles.

### Final Answer for problem 4:
The quadrilateral with vertices $ D(-3,3) $, $ E(-1,1) $, $ F(1,-4) $, $ G(-3,0) $ is a trapezoid (because $ DE\parallel FG $) and it is not isosceles (because the legs $ EF $ and $ GD $ are not congruent).

Let's solve problem 5: $ M(-2, 0) $, $ N(0, 4) $, $ P(5, 4) $, $ Q(8, 0) $

### Step 1: Calculate the slopes of the sides
- Slope of $ MN $: For $ M(-2,0) $ and $ N(0,4) $, $ m_{MN}=\frac{4 - 0}{0 + 2}=\frac{4}{2}=2 $
- Slope of $ NP $: For $ N(0,4) $ and $ P(5,4) $, $ m_{NP}=\frac{4 - 4}{5 - 0}=\frac{0}{5}=0 $
- Slope of $ PQ $: For $ P(5,4) $ and $ Q(8,0) $, $ m_{PQ}=\frac{0 - 4}{8 - 5}=\frac{-4}{3}=-\frac{4}{3} $
- Slope of $ QM $: For $ Q(8,0) $ and $ M(-2,0) $, $ m_{QM}=\frac{0 - 0}{-2 - 8}=\frac{0}{-10}=0 $

Since $ m_{NP}=m_{QM}=0 $, $ NP\parallel QM $. So, the quadrilateral $ MNPQ $ is a trapezoid.

### Step 2: Check if it is isosceles (check the lengths of the legs $ MN $ and $ PQ $)
- Length of $ MN $: For $ M(-2,0) $ and $ N(0,4) $, $ d_{MN}=\sqrt{(0 + 2)^2+(4 - 0)^2}=\sqrt{2^2+4^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5} $
- Length of $ PQ $: For $ P(5,4) $ and $ Q(8,0) $, $ d_{PQ}=\sqrt{(8 - 5)^2+(0 - 4)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5 $

Wait, we made a mistake. Let's re - calculate the length of $ MN $: $ d_{MN}=\sqrt{(0 - (-2))^2+(4 - 0)^2}=\sqrt{(2)^2 + 4^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.47 $, and $ d_{PQ}=\sqrt{(8 - 5)^2+(0 - 4)^2}=\sqrt{9 + 16}=\sqrt{25}=5 $. Wait, actually, we should check the lengths of the non - parallel sides. Wait, $ NP $ and $ QM $ are the parallel sides (horizontal lines). The legs are $ MN $ and $ PQ $. Wait, no, in a trapezoid, the legs are the non - parallel sides. Wait, $ MN $ has slope 2, $ PQ $ has slope $-\frac{4}{3}$, $ NP $ and $ QM $ have slope 0 (parallel). So the legs are $ MN $ and $ PQ $. But wait, let's check the lengths of the other pair of sides? Wait, no, the definition of an isosceles trapezoid is that the non - parallel sides (legs) are congruent. Wait, but maybe we made a mistake in identifying the legs. Wait, actually, in a trapezoid, the two parallel sides are called bases, and the other two are legs. So bases are $ NP $ and $ QM $ (slope 0). Legs are $ MN $ and $ PQ $. But let's check the lengths of the segments $ MP $ and $ NQ $? No, wait, the correct way is: For a trapezoid with bases $ b_1 $ and $ b_2 $, the legs are the non - parallel sides. To check if it is isosceles, we can also check if the distances from the endpoints of one base to the endpoints of the other base are equal.

Let's calculate the length of $ MQ $ and $ NP $? No, $ NP $ is a base. Wait, $ M(-2,0) $, $ Q(8,0) $, length $ MQ=\sqrt{(8 + 2)^2+(0 - 0)^2}=10 $. $ N(0,4) $, $ P(5,4) $, length $ NP = 5 $. Wait, no, the formula for isosceles trapezoid: the legs are congruent, or the diagonals are congruent. Let's calculate the diagonals.

Diagonal $ MP $: For $ M(-2,0) $ and $ P(5,4) $, $ d_{MP}=\sqrt{(5 + 2)^2+(4 - 0)^2}=\sqrt{49 + 16}=\sqrt{65} $

Diagonal $ NQ $: For $ N(0,4) $ and $ Q(8,0) $, $ d_{NQ}=\sqrt{(8 - 0)^2+(0 - 4)^2}=\sqrt{64 + 16}=\sqrt{80}=4\sqrt{5}\approx8.94 $

Wait, that's not equal. Wait, we made a mistake in slope calculation. Wait, $ MN $: $ M(-2,0) $, $ N(0,4) $, slope $ m=\frac{4 - 0}{0+2}=2 $. $ PQ $: $ P(5,4) $, $ Q(8,0) $, slope $ m=\frac{0 - 4}{8 - 5}=-\frac{4}{3} $. $ NP $: $ N(0,4) $, $ P(5,4) $, slope 0. $ QM $: $ Q(8,0) $, $ M(-2,0) $, slope 0. So $ NP\parallel QM $. Now, let's calculate the lengths of the legs $ MN $ and $ PQ $ again.

$ d_{MN}=\sqrt{(0 + 2)^2+(4 - 0)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\approx4.47 $

$ d_{PQ}=\sqrt{(8 - 5)^2+(0 - 4)^2}=\sqrt{9 + 16}=\sqrt{25}=5 $

Wait, they are not equal. But wait, let's check the lengths of $ MQ $ and $ NP $? No, $ MQ $ is a base? No, $ MQ $ is not a base. Wait, the bases are $ NP $ (length 5) and $ QM $ (length 10). Wait, maybe we misidentified the bases. Wait, a trapezoid has at least one pair of parallel sides. We have two pairs? $ NP $ (slope 0) and $ QM $ (slope 0) are parallel, $ MN $ has slope 2, $ PQ $ has slope $-\frac{4}{3}$. Wait, no, if two pairs of sides are parallel, it's a parallelogram. But $ MN $ and $ PQ $ are not parallel (slopes 2 and $-\frac{4}{3}$ are not equal). So it's a trapezoid (only one pair of parallel sides: $ NP\parallel QM $).

Now, to check if it's isosceles, we can check the diagonals. Diagonal $ MP $: $ M(-2,0) $ to $ P(5,4) $: $ \sqrt{(5 + 2)^2+(4 - 0)^2}=\sqrt{49 + 16}=\sqrt{65} $

Diagonal $ NQ $: $ N(0,4) $ to $ Q(8,0) $: $ \sqrt{(8 - 0)^2+(0 - 4)^2}=\sqrt{64 + 16}=\sqrt{80} $

Since $ \sqrt{65}
eq\sqrt{80} $, the diagonals are not equal. Wait, but maybe we made a mistake. Wait, let's re - calculate the coordinates. $ M(-2,0) $, $ N(0,4) $, $ P(5,4) $, $ Q(8,0) $. Let's plot these points mentally. $ N(0,4) $ and $ P(5,4) $ are on the horizontal line $ y = 4 $. $ M(-2,0) $ and $ Q(8,0) $ are on the horizontal line $ y = 0 $. So it's a trapezoid with bases $ NP = 5 $ and $ MQ=10 $. The legs are $ MN $ and $ PQ $. Now, let's calculate the length of $ MN $: $ \sqrt{(0 + 2)^2+(4 - 0)^2}=\sqrt{4 + 16}=\sqrt{20} $, length of $ PQ $: $ \sqrt{(8 - 5)^2+(0 - 4)^2}=\sqrt{9 + 16}=\sqrt{25} $. They are not equal. But wait, maybe the problem is that we have a trapezoid and we can check the other pair of sides. Wait, no, the definition is at least one pair of parallel sides. But in this case, we have one pair. Wait, maybe I made a mistake in the slope of $ MN $ and $ PQ $. Wait, $ MN $: from $ (-2,0) $ to $ (0,4) $, change in $ x = 2 $, change in $ y = 4 $, slope $ 4/2 = 2 $. $ PQ $: from $ (5,4) $ to $ (8,0) $, change in $ x = 3 $, change in $ y=-4 $, slope $ - 4/3 $. Not parallel. So it's a trapezoid. Now, to check if it's isosceles, the legs must be congruent. Since $ \sqrt{20}
eq5 $, it's not isosceles? Wait, but wait, let's calculate the distance between $ M $ and $ P $ and $ N $ and $ Q $ again. $ M(-2,0) $, $ P(5,4) $: $ \sqrt{(5 + 2)^2+(4 - 0)^2}=\sqrt{49 + 16}=\sqrt{65} $. $ N(0,4) $, $ Q(8,0) $: $ \sqrt{(8 - 0)^2+(0 - 4)^2}=\sqrt{64 + 16}=\sqrt{80} $. Not equal.

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QUESTION IMAGE

Question

Explanation:

Step 1: Recall the definition of a trapezoid and isosceles trapezoid

Step 2: Calculate the slopes of the sides

Step 3: Check if it is isosceles (check the lengths of the legs \( XY \) and \( ZW \))

Final Answer for problem 3:

Step 1: Calculate the slopes of the sides

Step 2: Check if it is isosceles (check the lengths of the legs \( EF \) and \( GD \))

Final Answer for problem 4:

Answer:

Step 1: Recall the definition of a trapezoid and isosceles trapezoid

Step 2: Calculate the slopes of the sides

Step 3: Check if it is isosceles (check the lengths of the legs \( XY \) and \( ZW \))

Final Answer for problem 3:

Step 1: Calculate the slopes of the sides

Step 2: Check if it is isosceles (check the lengths of the legs \( EF \) and \( GD \))

Final Answer for problem 4:

Step 1: Calculate the slopes of the sides

Step 2: Check if it is isosceles (check the lengths of the legs \( MN \) and \( PQ \))