Exercises: Use distance formula for area - Common Core Math - HS Geometry

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

What is the distance between the points $(1, 2)$ and $(4, 6)$ ?

A.

3

B.

7

C.

$5$

D.

$\sqrt{7}$

2.

A rectangle has vertices at $A(0, 0)$ , $B(6, 0)$ , $C(6, 4)$ , and $D(0, 4)$ .
What is its area in square units?

3.

A triangle has a horizontal base of length 8 units and a vertical height of
5 units from the base to the opposite vertex. What is its area?

A.

40 square units

B.

$20$ square units

C.

13 square units

D.

$\sqrt{89}$ square units

B

Fluency Practice

Apply the distance formula and area formulas to each figure.

1.

Triangle $ABC$ has vertices $A(0, 0)$ , $B(5, 0)$ , and $C(5, 12)$ .
What is the perimeter of triangle $ABC$ ?

2.

Quadrilateral $PQRS$ has vertices $P(0, 0)$ , $Q(4, 0)$ , $R(6, 3)$ ,
and $S(2, 3)$ . Compute the exact perimeter of $PQRS$ .

3.

Triangle $DEF$ has vertices $D(1, 1)$ , $E(5, 1)$ , and $F(3, 4)$ .
A student says the perimeter is $4 + \sqrt{13}$ units.
What error did the student make, and what is the correct perimeter?

A.

The student forgot side $FD$ . The correct perimeter is $4 + 2\sqrt{13}$ units.

B.

The student computed $DE$ incorrectly. The correct perimeter is $4 + 2\sqrt{13}$ units
but for a different reason.

C.

The student used the wrong formula. The correct perimeter is $4 + \sqrt{13}$ units.

D.

No error was made; $4 + \sqrt{13}$ is correct.

4.

Rectangle $ABCD$ has vertices $A(0, 0)$ , $B(4, 2)$ , $C(3, 4)$ , and $D(-1, 2)$ .
The adjacent sides are perpendicular. What is the area of rectangle $ABCD$ ?

5.

Triangle $PQR$ has vertices $P(1, 2)$ , $Q(7, 2)$ , and $R(4, 8)$ .
What is the area of triangle $PQR$ ?

C

Mixed Practice

Apply the appropriate method for each problem.

1.

Rectangle $KLMN$ has vertices $K(2, 1)$ , $L(8, 1)$ , $M(8, 5)$ , and $N(2, 5)$ .
Which expression gives the correct area?

A.

$(8 - 2) + (5 - 1) = 10$ square units

B.

$(8 - 2) \times (5 - 1) = 24$ square units

C.

$\sqrt{(8-2)^2 + (5-1)^2} = \sqrt{52}$ square units

D.

$2(8 - 2) + 2(5 - 1) = 20$ square units

2.

Triangle $ABC$ has vertices $A(0, 0)$ , $B(6, 2)$ , and $C(2, 5)$ .
Use the coordinate area formula
$\text{Area} = \tfrac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$
to find the area of triangle $ABC$ .

3.

Triangle $PQR$ has vertices $P(-2, 3)$ , $Q(4, -1)$ , and $R(1, 5)$ .
Use the coordinate area formula to find the area.

4.

Quadrilateral $ABCD$ has vertices $A(0, 0)$ , $B(4, 0)$ , $C(5, 3)$ , and $D(1, 4)$
listed in order. Apply the Shoelace formula to find the area of $ABCD$ .

D

Word Problems

Read each scenario carefully. Choose the appropriate method and show your work.

1.

A farmer wants to fence a triangular plot of land. The corners of the plot
are located at coordinates $A(0, 0)$ , $B(8, 0)$ , and $C(3, 6)$ on a map
where each unit represents 1 meter.

How many meters of fencing does the farmer need? Round your answer to the
nearest tenth of a meter.

2.

An architect's floor plan places a rectangular room with corners at
$J(1, 2)$ , $K(7, 2)$ , $L(7, 6)$ , and $M(1, 6)$ , where each unit
represents 1 foot.

What is the area of the room in square feet?

3.

A surveyor maps a triangular lot with corners at $(0, 0)$ , $(80, 0)$ ,
and $(30, 60)$ , where each unit represents 1 foot.

What is the area of the lot in square feet?

4.

A pond is approximated by a pentagon with vertices
$(2, 1)$ , $(5, 0)$ , $(8, 2)$ , $(7, 5)$ , and $(3, 6)$ , where each
unit represents 1 meter.

Use the Shoelace formula to find the area of the pond in square meters.

E

Error Analysis

A student's work is shown below. Identify the error and explain how to fix it.

1.

Maya is computing the perimeter of quadrilateral $ABCD$ with vertices
$A(0, 0)$ , $B(3, 0)$ , $C(5, 4)$ , and $D(1, 4)$ .

Maya's work:

$AB = 3 - 0 = 3$
$BC = \sqrt{(5-3)^2 + (4-0)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}$
$CD = 5 - 1 = 4$
Perimeter $= 3 + 2\sqrt{5} + 4 = 7 + 2\sqrt{5} \approx 11.47$ units

Identify all errors in Maya's work and compute the correct perimeter.

A.

Maya forgot side $DA$ . She also used only the x-difference for $CD$ instead of
the full distance formula (though $CD$ is horizontal so subtracting x-coordinates
is valid here). The correct perimeter is $7 + 2\sqrt{5} + \sqrt{2} \approx 12.9$ units.

B.

Maya computed $BC$ incorrectly. All other steps are correct.

C.

Maya's work is entirely correct.

D.

Maya should have used the Shoelace formula, not the distance formula.

2.

Kai is computing the area of triangle $XYZ$ with vertices $X(1, 4)$ , $Y(5, 0)$ ,
and $Z(3, 6)$ using the coordinate area formula.

Kai's work:

= \frac{1}{2}[-6 + 10 + 12] = \frac{1}{2}(16) = 8 \text{ square units}$$

Identify the error in Kai's work and compute the correct area.

A.

Kai's arithmetic inside the brackets is wrong.

B.

Kai omitted the absolute value. The expression inside the brackets is $-6 + 10 + 12 = 16$ ,
which is positive here, so the answer 8 is actually correct in this case.

C.

Kai should have reported a negative area: $-8$ square units.

D.

Kai used the wrong formula — the Shoelace formula is required for triangles.

F

Challenge

These problems require multiple steps and careful reasoning.

1.

Triangle $MNP$ has vertices $M(0, 0)$ , $N(5, 1)$ , and $P(2, 4)$ .
Compute the area using the coordinate area formula and verify it using
the bounding rectangle method (enclose the triangle in the smallest
axis-aligned rectangle and subtract the three surrounding right triangles).
Report the area in square units.

2.