Algebra for Geometry

In one sentence

A length in a geometry problem that you “cannot see” can be given a name first: call it $x$ .

Then let the figure write the equation for you. Every theorem in the figure is a source of equations:

right triangle → the Pythagorean theorem $a^{2} + b^{2} = c^{2}$
similar triangles → a proportion
circle → equal tangent lengths, $P A \cdot P B = P T^{2}$
the same area computed twice → an equation

Write the equation, solve it, then check it in the figure (is the length positive? is the point still on the segment?). If the question asks for the “largest” or “smallest”, write that quantity as a function of $x$ and hand it to completing the square.

Predict first

In the rectangle above, $A D = 15$ and $A B = 9$ . Drag $E$ around: roughly what is $D E$ when $D$ folds exactly onto $B C$ ?

Choose an answer first; then you can see the result.

Stop 1 · A fold gives a Pythagorean equation (Vol. 4 · Right triangles)

In rectangle $A B C D$ , $A D = 15$ and $A B = 9$ . $E$ is on $C D$ . Folding along $A E$ puts $D$ exactly onto the point $F$ on side $B C$ . Find $D E$ .

Example: folding a corner

Step 1 / 5

Give it a name
Let $D E = x$ . Folding does not change lengths, so $E F = x$ and $A F = A D = 15$ . Also $E C = 9 - x$ .

$D E = E F = x, E C = 9 - x$
Work out what you can
Let Pythagoras write the equation
Solve it
Look back at the figure

Stop 2 · Hand the maximum to a quadratic function (Vol. 3 · Applications of functions)

$△ A B C$ has base $B C = 12$ and height $8$ . A rectangle is placed inside it with one side on $B C$ and its other two vertices on $A B$ and $A C$ . What is the largest possible area of the rectangle?

Predict first

What height $x$ do you think gives the rectangle the largest area?

Choose an answer first; then you can see the result.

Example: a rectangle in a triangle

Step 1 / 5

Call the height x
The rectangle has height $x$ . Its top side cuts off a small triangle of height $8 - x$ , similar to the big triangle.

$\frac{width}{12} = \frac{8 - x}{8}$
Write the width in terms of x too
The area is a quadratic function of x
Complete the square
Look back at the figure

Stop 3 · Tangent lengths give a system of equations (Vol. 5 · Circumcenter and incenter)

$△ A B C$ has sides $A B = 13$ , $B C = 14$ and $C A = 15$ . The incircle splits each side into two parts. How long is each part?

Example: the incircle of 13, 14, 15

Step 1 / 5

Six segments, three lengths
The two tangents from a point outside a circle have equal length. So the two parts starting at $A$ are equal, and so are the two at $B$ and the two at $C$ .

$A F = A E, B F = B D, C D = C E$
Three names
Add all three
Subtract to get each one
The radius comes for free

Stop 4 · The power of a point gives a quadratic equation (Vol. 5 · Power of a point)

From a point $P$ outside a circle, draw the tangent $P T$ with $P T = 6$ . A secant through $P$ meets the circle at $A$ and $B$ ( $A$ is closer to $P$ ), and $A B = 5$ . Find $P A$ .

Example: a tangent and a secant

Step 1 / 4

Let PA = x
Then $P B = P A + A B = x + 5$ .

$P A = x, P B = x + 5, P T = 6$
The power of a point gives an equation
Solve the quadratic
Check it in the figure

Practice

Exercise 1Basic

One leg of a right triangle is $6$ , and the hypotenuse is $2$ longer than the other leg. How long is the other leg?

Exercise 2Advanced

In $△ A B C$ , $A B = 7$ , $A C = 8$ and $B C = 9$ . The incircle touches $A B$ at $F$ . Find $A F$ .

Exercise 3Challenge

A right triangle has legs $6$ and $8$ . A rectangle is placed inside it: one vertex is the right-angle vertex, two sides lie along the legs, and the opposite vertex is on the hypotenuse. What is the largest possible area of the rectangle?

In one sentence

Stop 1 · A fold gives a Pythagorean equation (Vol. 4 · Right triangles)

Example: folding a corner

Stop 2 · Hand the maximum to a quadratic function (Vol. 3 · Applications of functions)

Example: a rectangle in a triangle

Stop 3 · Tangent lengths give a system of equations (Vol. 5 · Circumcenter and incenter)

Example: the incircle of 13, 14, 15

Stop 4 · The power of a point gives a quadratic equation (Vol. 5 · Power of a point)

Example: a tangent and a secant

Practice

Mistake Museum · Exhibits from this stop

Keeping both roots

Ignoring the range of x when finding a maximum

Stops on this line