Similarity and Dilations: math practice, Grade 10 – download the PDF

1 Dilating points from the origin ★★★

Find the image of each point under the dilation centered at the origin with scale factor \(k=3\).

1. \(A(2,-1)\)
2. \(B(-4,5)\)
3. \(C(0,6)\)

2 Finding the scale factor ★★★

A triangle with sides 4 in., 6 in. and 8 in. is dilated to a triangle with sides 10 in., 15 in. and 20 in.

1. What is the scale factor?
2. What scale factor maps the large triangle back to the small one?

3 True or false? ★★★

Decide whether each statement is true or false and justify.

1. Any two squares are similar.
2. Any two rectangles are similar.
3. Any two equilateral triangles are similar.
4. A dilation with \(k=0.5\) makes a figure smaller.

4 Corresponding sides ★★★

\(\triangle ABC\sim\triangle DEF\) with \(AB=6\), \(BC=9\), \(AC=12\) and \(DE=8\). Find \(EF\) and \(DF\).

5 Enlarging a photo ★★★

A photo is 4 in. wide and 6 in. tall. It is enlarged, keeping its shape, so that it is 10 in. wide. How tall is the enlargement? What is the scale factor?

6 Reading a map scale ★★★

On a map, 1 inch represents 15 miles.

1. Two towns are 3.5 in. apart on the map. What is the real distance?
2. Two lakes are 120 miles apart. How far apart are they on the map?

7 Angle sum and AA ★★★

Triangle \(ABC\) has \(\angle A=52^\circ\) and \(\angle B=71^\circ\). Triangle \(DEF\) has \(\angle D=71^\circ\) and \(\angle F=57^\circ\). Are the triangles similar? Which angles match?

8 Shadows and height ★★★

At the same moment, a 1.8 m person casts a 2.4 m shadow and a tree casts a 14.4 m shadow.

Explain why the two triangles are similar and find the height \(h\) of the tree.

9 SSS or not? ★★★

For each pair, decide whether the triangles are similar. If so, give the scale factor.

1. 6, 8, 10 and 9, 12, 15
2. 5, 7, 9 and 10, 14, 19

10 Using SAS ★★★

In triangle \(ABC\), \(AB=6\), \(AC=9\) and \(\angle A=40^\circ\). In triangle \(DEF\), \(DE=10\), \(DF=15\) and \(\angle D=40^\circ\).

1. Show that the triangles are similar.
2. If \(BC=7.2\), find \(EF\).

11 A line parallel to a side ★★★

In the figure, \(DE\parallel BC\), \(AD=6\), \(DB=4\) and \(AE=9\). Find \(EC\) and \(AC\).

12 Is it parallel? ★★★

In triangle \(ABC\), \(D\) is on \(AB\) and \(E\) is on \(AC\). Decide whether \(DE\parallel BC\).

1. \(AD=5\), \(DB=10\), \(AE=4\), \(EC=8\)
2. \(AD=3\), \(DB=5\), \(AE=4\), \(EC=6\)

13 Bisecting an angle ★★★

In triangle \(ABC\), \(AB=12\), \(AC=20\) and \(BC=24\). The bisector of \(\angle A\) meets \(BC\) at \(D\). Find \(BD\) and \(DC\).

14 Perimeter and area of a model ★★★

A large polygon is an enlargement of a small, similar polygon with scale factor 3. The small polygon has perimeter 14 cm and area 5 cm². Find the perimeter and the area of the large polygon.

15 Altitude on the hypotenuse ★★★

Triangle \(ABC\) has a right angle at \(C\), with \(AC=15\), \(BC=20\) and \(AB=25\). Point \(D\) is the foot of the altitude from \(C\) to \(AB\).

1. Prove that \(\triangle ACD\sim\triangle ABC\).
2. Find \(AD\) and \(CD\).

16 Diagonals of a trapezoid ★★★

In trapezoid \(ABCD\), \(AB\parallel CD\), \(AB=12\) and \(CD=8\). The diagonals meet at \(E\) and \(AE=9\).

1. Prove that \(\triangle ABE\sim\triangle CDE\).
2. Find \(CE\) and \(AC\).

17 Areas of similar triangles ★★★

Two similar triangles have areas 18 cm² and 50 cm². The smaller one has perimeter 21 cm.

1. Find the scale factor from the smaller to the larger triangle.
2. Find the perimeter of the larger triangle.

18 Scale model of a building ★★★

An architect builds a model at the scale 1 in. : 4 ft (so real lengths are 48 times the model lengths). The model is 9.5 in. tall and its front wall has an area of 150 in².

1. How tall is the real building in feet?
2. What is the real area of the front wall in square feet?

19 Dilation with another center ★★★

The center of a dilation is \(O(1,2)\). Use \(P'=(a+k(x-a),\,b+k(y-b))\) to find the image of each point.

1. \(A(3,4)\) with \(k=2\)
2. \(B(0,0)\) with \(k=2\)
3. \(C(5,-2)\) with \(k=\tfrac12\)

20 Solving with an unknown ★★★

In triangle \(ABC\), \(DE\parallel BC\) with \(D\) on \(AB\) and \(E\) on \(AC\). Given \(AD=x\), \(DB=6\), \(AE=x+1\) and \(EC=9\), find \(x\), \(AD\) and \(AE\).

21 Proving the theorem ★★★

In triangle \(ABC\), \(D\) lies on \(AB\) and \(E\) lies on \(AC\) with \(DE\parallel BC\). Prove that \(\triangle ADE\sim\triangle ABC\) and explain why \(\dfrac{AD}{AB}=\dfrac{DE}{BC}\).