Triangle ∆ABC is similar to ∆DEF. AB is 8 inches
long, BC is 10 inches long, and AC is 16 inches long.
If the longest side of ∆DEF is 40 inches long, what is
the perimeter, in inches, of ∆DEF ?
A.
B.
C.
D.
E.
ACT Practice
Plane Geometry
E
74
B
85
8
10
90
136
A
C D
F
16
40
170
If the two triangles are similar, then the corresponding sides are in proportion
AB AC {corresponding sides
=
DE DF are proportional}
8
16 {substituted given sides
=
DE 40 into proportion}
16x = 320 {cross-multiplied}
BC AC
=
EF DF
10 16
=
EF 40
{substituted given sides
into proportion}
16x = 400
{cross-multiplied}
DE = 20
EF = 25
{divided each side by 16}
{corresponding sides
are proportional}
{divided each side by 16}
Perimeter of ∆DEF = 20 + 25 + 40 = 85 inches
© Mr. Sims
ACT Practice
Plane Geometry
3
3
Shaded region consists of 4 equal triangles.
Area of triangle =
=
Area of 1 triangle =
base x height
2
3(3)
{substituted into area formula}
2
9
{multiplied 3 by 3 in numerator}
2
Area of all 4 triangles:
=4x
=
9
9
{one triangle’s area is }
2
2
36
{multiplied}
2
= 18
{divided 36 by 2}
© Mr. Sims
ACT Practice
Plane Geometry
∠ABC is a right triangle {∠B inscribes a semicircle}
∠𝐀𝐁𝐂 = 𝟗𝟎° {a right triangle measures 90°}
Also:
AB ≅ BC {given}
∠A ≅ ∠C {∆ABC is an isosceles triangle}
∠A = 45° and ∠C = 45° {angles of triangle
add up to 180°}
© Mr. Sims
In the standard (x,y) coordinate plane, what are the
coordinates of the midpoint of a line segment with
endpoints (-1,3) and (2,7) ?
F.
1
,5
2
G.
9
1,
2
H.
3
,2
2
J. (1,4)
K. (3,2)
Midpoint =
x1 +x2 y1 +y2
,
2
2
x1,y1
ACT Practice
Coordinate Geometry
{the two points have coordinates of (x1,y1) and (x2,y2)}
x2,y2
(-1,3) and (2,7)
M=
−1+2 3+7
,
2
2
M=
1 10
,
2 2
M=
𝟏
,𝟓
𝟐
{substituted coordinates into midpoint formula}
{added in numerators}
{divided 10 by 2}
© Mr. Sims
ACT Practice
Elementary Algebra
In a downhill ski race, Margo posted a time of
2 minutes and 24 seconds for a course 1.2 miles long.
About how many miles per hour did she average
for the race?
1.2 miles
she went
2 minutes and 24 seconds
A. 60
B. 30 change 2 minutes into hours
change 24 seconds into hours
C. 20 = 2 ÷ 60 {sixty minutes in 1 hour}
= 24 ÷ 3600 {3,600 seconds in 1 hour}
D. 3
24
2
E. 2
=
{wrote as a fraction}
=
{wrote as a fraction}
3600
60
=
1
30
=
5
150
1
30
hours
{reduced}
+
1
{added the two hour totals}
150
+
1
{got common denominator}
150
=
=
6
{combined numerators over common denominator}
150
=
1
hours {reduced}
25
1
150
hours
{reduced}
Divide miles by hours
1
= 1.2 miles ÷
hours
25
= 1.2 x 25 {multiplied by reciprocal}
= 30 miles per hour {multiplied}
© Mr. Sims
ACT Practice
Intermediate Algebra
When x = 3, f(x) is 2
therefore, g(f(3)) is really g(2)
which equals -3
© Mr. Sims
For positive real numbers x, y, and z, which of the
1
2
2
3
ACT Practice
Intermediate Algebra
5
6
following expressions is equivalent to x y z ?
A.
B.
C.
D.
E.
3
6
xy²z³
Change all exponents to have same denominator
x y2z5
6
x3y2z5
6
x3y4z5
11
xy 2 z 5
1
2
2
3
5
6
3
6
4
6
5
6
x y z
= x y z
=
𝟔
{changed to denominator of 6}
𝐱𝟑𝐲𝟒𝐳𝟓
With a fractional exponent:
- the denominator is the “root”
- the numerator is the new exponent when under the radical sign
© Mr. Sims
1
A formula for the area of a rhombus is A = d1 d2,
2
where d1 and d2 are the lengths of the diagonals. Which
of the following is an expression for d2 ?
F.
G.
H.
2A
d1
ACT Practice
Intermediate Algebra
Solve the equation for d2
1
A = d1 d 2
2
A
2d1
2A = d1d2
Ad1
2
𝟐𝐀
𝐝𝟏
= 𝐝𝟐
{multiplied each side by 2}
{divided each side by d1}
J. 2(A – d1)
K. A −
d1
2
© Mr. Sims
ACT Practice
Pre-Algebra
Vehicle A
1,008 miles ÷ 14 mpg
= 72 gallons used
Vehicle B
1,008 miles ÷ 36 mpg
= 28 gallons used
Difference in gallons
= 72 – 28
= 44 gallons more for vehicle A
© Mr. Sims
ACT Practice
Elementary Algebra
= -81t2 + t + 54
combine like terms
© Mr. Sims
ACT Practice
Plane Geometry
6
6
In an equilateral triangle, all sides are equal
6
6
6
In a square, all sides are equal
Perimeter is the distance around the outside
Perimeter of ABCDE
= 6 + 6 + 6 + 6 + 6 {added all sides of ABCDE, not including 𝑬𝑩}
= 30
© Mr. Sims