AMC 10 · 2003 · #6
Grade 8 geometry-2dMany television screens are rectangles that are measured by the length of their diagonals. The ratio of the horizontal length to the height in a standard television screen is 4:3. The horizontal length of a "27-inch" television screen is closest, in inches, to which of the following?
Pick an answer.
AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A television screen is a rectangle whose horizontal length and height are in the ratio $4:3$. The screen is called "$27$-inch" because its diagonal measures $27$ inches. Find which answer choice is closest to the horizontal length, in inches.
Givens: The screen is a rectangle; Horizontal length : height $= 4:3$; The diagonal is $27$ inches (that is what "$27$-inch" means); Answer choices: (A) $20$, (B) $20.5$, (C) $21$, (D) $21.5$, (E) $22$
Unknowns: The horizontal length of the screen, in inches
Understand
Restated: A television screen is a rectangle whose horizontal length and height are in the ratio $4:3$. The screen is called "$27$-inch" because its diagonal measures $27$ inches. Find which answer choice is closest to the horizontal length, in inches.
Givens: The screen is a rectangle; Horizontal length : height $= 4:3$; The diagonal is $27$ inches (that is what "$27$-inch" means); Answer choices: (A) $20$, (B) $20.5$, (C) $21$, (D) $21.5$, (E) $22$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #4 Introduce a Variable, #5 Look for a Pattern
The words "horizontal," "height," and "diagonal" describe a shape, so Tool #1 (Draw a Diagram) comes first: drawing the diagonal splits the rectangle into a right triangle whose legs are the length and the height and whose hypotenuse is the $27$-inch diagonal. Tool #4 (Introduce a Variable) turns the ratio $4:3$ into legs $4x$ and $3x$, so one unknown $x$ describes both sides at once. Tool #5 (Look for a Pattern) spots that legs in the ratio $3:4$ make a $3$-$4$-$5$ right triangle, so the hypotenuse is $5x$ — no square roots needed, just solve $5x=27$.
Execute — Answer: D
4.G.A.2 Step 1 Draw the diagonal, make a right triangle
- Draw the rectangle and its diagonal.
- The diagonal cuts the rectangle into two right triangles.
- In one of them, one leg is the horizontal length, the other leg is the height, and the hypotenuse is the diagonal, which is $27$ inches.
💡 The diagonal of a rectangle is always the hypotenuse of a right triangle whose legs are the two sides.
6.RP.A.3 Step 2 Name the sides with the ratio
- The length and height are in the ratio $4:3$, so write the length as $4x$ and the height as $3x$ for some positive number $x$.
- Now both sides are described by the single unknown $x$.
💡 A ratio $4:3$ means both sides grow together, so one multiplier $x$ fixes both.
8.G.B.7 Step 3 Use the $3$-$4$-$5$ triangle
- Legs of $3x$ and $4x$ form a $3$-$4$-$5$ right triangle scaled by $x$, so the hypotenuse is $5x$.
- Checking with the Pythagorean Theorem: $(3x)^2+(4x)^2=9x^2+16x^2=25x^2=(5x)^2$.
- The hypotenuse is the diagonal, so $5x=27$.
💡 Legs in the ratio $3:4$ always give a hypotenuse $5$ parts long, the classic $3$-$4$-$5$ triangle.
6.RP.A.3 Step 4 Solve for the length
- From $5x=27$, divide both sides by $5$ to get $x=\frac{27}{5}=5.4$.
- The horizontal length is $4x=4\times5.4=21.6$ inches.
- Among the choices, $21.6$ is closest to $21.5$, so the answer is (D).
💡 Once $x$ is known, the length is just four of those equal parts.
4.G.A.2 Draw the rectangle and its diagonal. The diagonal cuts the rectangle into two ri 6.RP.A.3 The length and height are in the ratio $4:3$, so write the length as $4x$ and th 8.G.B.7 Legs of $3x$ and $4x$ form a $3$-$4$-$5$ right triangle scaled by $x$, so the hy 6.RP.A.3 From $5x=27$, divide both sides by $5$ to get $x=\frac{27}{5}=5.4$. The horizont Review
Reasonableness: The horizontal length must be less than the diagonal of $27$ inches but more than the height, so a value in the low $20$s is sensible — $21.6$ fits. A quick sanity check: the height is $3x=16.2$ inches, and $21.6^2+16.2^2=466.56+262.44=729=27^2$, so the sides really do give a $27$-inch diagonal. The trap is treating $27$ as a side instead of the diagonal; then $4$ of $7$ parts of $27$ would be about $15.4$, which is not even a choice, confirming the diagonal reading is right.
Alternative: Tool #4 (Introduce a Variable) without spotting the $3$-$4$-$5$ shortcut: set length $=4x$, height $=3x$, and apply the Pythagorean Theorem directly. Then $(4x)^2+(3x)^2=27^2$ gives $25x^2=729$, so $x^2=29.16$ and $x=5.4$. The length $4x=21.6\approx21.5$, the same choice (D).
CCSS standards used (min grade 8)
4.G.A.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size; recognize right triangles (Seeing that the diagonal splits the rectangle into a right triangle with the two sides as legs.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Turning the $4:3$ ratio into sides $4x$ and $3x$ and then computing the length $4x$.)8.G.B.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems (Relating the two legs to the $27$-inch hypotenuse to solve for $x$.)
⭐ The diagonal of a $4:3$ screen makes a $3$-$4$-$5$ triangle, so the diagonal is $5$ parts — split it into $5$ and take $4$ of them for the width.
⭐ The diagonal of a $4:3$ screen makes a $3$-$4$-$5$ triangle, so the diagonal is $5$ parts — split it into $5$ and take $4$ of them for the width.
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