Wize High School Grade 9 Math Textbook > 2D Geometry - Properties of 2D Figures

Similar Triangles
Similar triangles are triangles that have the same shape but different sizes.

How can we tell if two triangles are similar?
Using angles: The corresponding angles (angles in the same relative positions) are all the same

Example: 
∠A=∠D\angle A=\angle D∠A=∠D, ∠B=∠E\angle B=\angle E∠B=∠E, ∠C=∠F\angle C=\angle F∠C=∠F
We say that △ABC ∼ △DEF\triangle ABC ~\sim~\triangle DEF△ABC ∼ △DEF.

Using side lengths: The corresponding sides (sides in the same relative positions) have the same proportions

Example: 
ABRP=BCPQ=CAQR=2\dfrac{AB}{RP}=\dfrac{BC}{PQ}=\dfrac{CA}{QR}=2RPAB​=PQBC​=QRCA​=2
We say that △ABC ∼ △RPQ\triangle ABC~\sim~\triangle RPQ△ABC ∼ △RPQ.

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Example: Similar Triangles
Determine if the following triangles are similar.

Using Angles
Triangle 1: The 2 given angles in the first triangle are 20°20\degree20° and 90°90\degree90°. Since the sum of the interior angles in a triangle is 180°180\degree180°, we know that the missing angle is 70°70\degree70°.
Triangle 2: The 2 given angles in the second triangle are 20°20\degree 20° and 90°90\degree 90°. Similar to the first triangle, we know that the missing angle is 70°70\degree70°.
Matching up the same angles, we see that △ABC ∼ △QPR\triangle ABC~\sim~\triangle QPR△ABC ∼ △QPR.


Using Sides
Triangle 1: Using Pythagorean's theorem, we see that BC≈9.75cmBC\approx 9.75cmBC≈9.75cm
Triangle 2: Using Pythagorean's theorem, we see that PR≈2.44PR\approx2.44PR≈2.44
Matching up the sides:
longest sides: ACQR=123=4\dfrac{AC}{QR}=\dfrac{12}{3}=4QRAC​=312​=4
medium sides: BCRP≈9.752.44≈4\dfrac{BC}{RP}\approx\dfrac{9.75}{2.44}\approx4RPBC​≈2.449.75​≈4
shortest sides: ABQP=71.75=4\dfrac{AB}{QP}=\dfrac{7}{1.75}=4QPAB​=1.757​=4
Since the corresponding side lengths have the same proportions, we see that △ABC ∼ △QPR\triangle ABC~\sim~\triangle QPR△ABC ∼ △QPR.

Practice: Similar Triangles
If △ABC\triangle ABC△ABC is similar to △DEF\triangle DEF△DEF, solve for the missing variables w,x,y,z,p,qw, x, y, z, p, qw,x,y,z,p,q.

p=

(enter your answer in cm, do not include units)

q=

(enter your answer in cm, do not include units)

w=

(enter your answer in degrees, do not include units)

x=

(enter your answer in degrees, do not include units)

y=

(enter your answer in degrees, do not include units)

z=

(enter your answer in degrees, do not include units)

I don't know

Practice: Similar Triangle
Identify the similar triangles in the following diagrams, then find the missing lengths.

\triangle ABE

is similar to

\triangle

_________

y=

I don't know

Practice: Similar Triangles
A flashlight that is placed on the ground is pointed at a brick wall that is 20m away. A basketball player who is 1.9m tall stands 4m in front of the flashlight. Determine the height of the shadow of the basketball player that is cast on the brickwall.

Enter the height of the shadow in m, do not include units.

I don't know

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Similar Polygons
Similar polygons are figures that have the same shape but different sizes.

How can we tell if two polygons are similar?
Using angles: The corresponding angles (angles in the same relative positions) are all the same

Example:

Using side lengths: The corresponding sides (sides in the same relative positions) have the same proportions

Example:

Practice: Similar Polygons
A swimming pool has a smaller quadrilateral in the center that does not hold water. Find the perimeter of the smaller center quadrilateral.

Enter the perimeter of the smaller center quadrilateral in m, do not include units.

I don't know