Exercise 43 Page 424

Recall the classification of quadrilaterals. You can begin by finding the slopes of the sides.

Isosceles trapezoid

Practice makes perfect

Let's plot the given points on a coordinate plane and graph the quadrilateral.

Let's now determine the most precise name for our quadrilateral.

Name

To determine the most precise name for our quadrilateral, let's review the classification of quadrilaterals.

Quadrilateral	Definition
Parallelogram	Both pairs of opposite sides are parallel
Rhombus	Parallelogram with four congruent sides
Rectangle	Parallelogram with four right angles
Square	Parallelogram with four congruent sides and four right angles
Trapezoid	Quadrilateral with exactly one pair of parallel sides
Isosceles Trapezoid	Trapezoid with legs that are congruent
Kite	Quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent

Now, let's find the slopes of the sides using the Slope Formula.

Side	Slope Formula	Simplified
Slope of $\Seg{HA}\text{:}$ $(\col{\N 9,10}),$ $(\colIII{2,10})$	$\dfrac{\colIII{10}-\col{10}}{\colIII{2}-(\col{\N 9})}$	$0$
Slope of $\Seg{AT}\text{:}$ $(\colIII{2,10}),$ $(\colIV{4, 4})$	$\dfrac{\colIV{4}-\colIII{10}}{\colIV{4}-\colIII{2}}$	$\N 3$
Slope of $\Seg{TW}\text{:}$ $(\colIV{4, 4}),$ $(\colVI{\N 11, 4})$	$\dfrac{\colVI{4}-\colIV{4}}{\colVI{\N11}-\colIV{4}}$	$0$
Slope of $\Seg{WH}\text{:}$ $(\colVI{\N 11, 4}),$ $(\col{\N 9,10})$	$\dfrac{\col{10}-\colVI{4}}{\col{\N 9}-(\colVI{\N 11})}$	$3$

We can tell that the quadrilateral has exactly one pair of parallel sides. Therefore, it is either a trapezoid or an isosceles trapezoid. To check, we can find the lengths of the legs of the trapezoid using the Distance Formula.

Side	Distance Formula	Simplified
Length of $\Seg{AT}\text{:}$ $(\colIII{2,10}),$ $(\colIV{4, 4})$	$\sqrt{(\colIV{4}-\colIII{2})^2+(\colIV{4}-\colIII{10})^2}$	$\sqrt{40}$
Length of $\Seg{WH}\text{:}$ $(\colVI{\N 11, 4}),$ $(\col{\N 9,10})$	$\sqrt{(\col{\N 9}-(\colVI{\N 11}))^2+(\col{10}-\colVI{4})^2}$	$\sqrt{40}$

The legs of our trapezoid are congruent. Therefore, the most precise name for this quadrilateral is an isosceles trapezoid.

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Exercises p.420-424

Exercise 43 Page 424

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