Triangular Banner: Similarity, Area, and Inscribed Circle
A graphic designer is enlarging a triangular school logo for a banner. On a coordinate grid measured in inches, the original logo has vertices P(0,0), Q(4,0), and R(0,3). The enlarged banner triangle has vertices P'(1,1), Q'(9,1), and R'(1,7). The designer claims the banner triangle is similar to the original and plans to cut a circular seal with a 4-inch diameter from the banner. The banner material is sold only in whole square feet. The designer also needs a coordinate rule that maps the original logo onto the banner triangle, because a printing program will use that rule to place smaller copies along the border. A coworker warns that the 4-inch seal may not fit and that the coordinate rule cannot be a similarity transformation because the triangle has been moved, not just stretched.
1. Show that the banner triangle is similar to the original triangle by comparing side lengths. State the scale factor and the ratio of their areas.
Read the worked answer
For the original triangle, (PQ=4), (PR=3), and (QR=\sqrt{4^2+3^2}=\sqrt{25}=5).
For the banner triangle, (P'Q'=9-1=8), (P'R'=7-1=6), and (Q'R'=\sqrt{8^2+6^2}=\sqrt{100}=10).
The side ratios are
[\frac{P'Q'}{PQ}=\frac{8}{4}=2,\quad \frac{P'R'}{PR}=\frac{6}{3}=2,\quad \frac{Q'R'}{QR}=\frac{10}{5}=2.]
All three pairs of corresponding sides are proportional, so the triangles are similar by SSS similarity. The scale factor is 2. The ratio of the areas is the square of the scale factor:
[2^2=4.]
Check: the original area is (\frac12(4)(3)=6) square inches, and the banner area is (\frac12(8)(6)=24) square inches, so (24/6=4).
2. Find a coordinate rule that maps the original triangle onto the banner triangle. Explain why it is a similarity transformation even though it includes a translation.
Read the worked answer
Test the rule ((x,y)\to(2x+1,2y+1)):
[P(0,0)\to(1,1)=P',] [Q(4,0)\to(9,1)=Q',] [R(0,3)\to(1,7)=R'.]
So the rule is ((x,y)\to(2x+1,2y+1)). This can be viewed as a dilation by a factor of 2 centered at the origin, followed by a translation 1 unit right and 1 unit up.
A dilation changes all distances by the same factor, 2. A translation preserves all distances. The composition therefore multiplies every distance by 2 and preserves angle measures, so it is a similarity transformation. The coworker is incorrect: a similarity transformation may include translations, rotations, or reflections as well as dilations. Moving the triangle does not prevent the transformation from being a similarity.
3. The largest circle that can fit inside a right triangle has radius \(r=\frac{\text{area}}{\text{semiperimeter}}\). Compute \(r\) for the banner triangle and determine whether the 4-inch-diameter seal fits. If it fits, where must its center be?
Read the worked answer
The banner triangle has sides 6, 8, and 10 inches. Its area is
[\frac12(6)(8)=24\text{ square inches}.]
The semiperimeter is
[\frac{6+8+10}{2}=12\text{ inches}.]
So the inradius is
[r=\frac{24}{12}=2\text{ inches}.]
The largest possible circle inside the triangle has diameter (2r=4) inches, so the 4-inch-diameter seal fits exactly, tangent to all three sides.
Because the right angle is at (P'(1,1)) and the legs run horizontally and vertically, the incenter is 2 inches right and 2 inches up from (P'):
[(1+2,1+2)=(3,3).]
A check: the hypotenuse through ((9,1)) and ((1,7)) has equation (3x+4y=31). The distance from ((3,3)) to this line is
[\frac{|3(3)+4(3)-31|}{\sqrt{3^2+4^2}}=\frac{10}{5}=2,]
which confirms the circle centered at ((3,3)) with radius 2 fits.