SSAT Middle Level

Segments & Lengths

Segment arithmetic, midpoints, bisectors, and line segment relationships — 1D geometry without angle measurement

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Imagine a giant, delicious hot dog. 🌭 If you cut it into two pieces, those two pieces still add up to the whole hot dog! That's exactly how "Segments & Lengths" work in geometry. A line segment is just a straight line that has a clear start and end. If you place a point in the middle, you break that segment into two smaller pieces. The Segment Addition Postulate is just a fancy math rule that says "the small pieces add up to the big piece."

When you're taking the SSAT, geometry might look intimidating, but Segments and Lengths are actually some of the most fun puzzles on the test. You don't have to worry about weird angles or 3D shapes here—it's all happening on one straight line! Imagine it like a treasure map. 🗺️ Point A is your start, Point C is the treasure, and Point B is a cool tree you pass along the way. If it takes 10 steps to reach the tree, and 5 steps from the tree to the treasure, how far did you walk? $10 + 5 = 15$ steps!

What if you chop that hot dog perfectly in half? That exact middle spot is called the midpoint. A midpoint or a "bisector" cuts a segment into two equal halves. It's like sharing a candy bar with your best friend—you both get the exact same amount! 🍫 On the test, just keep your eyes peeled for those magic words, and remember: piece plus piece equals the whole thing!

Key Formula

If point B is between point A and point C, then

A B + B C = A C

. If M is the midpoint of AB, then

A M = M B = \frac{1}{2} A B

Practice Questions

3 practice questions for SSAT Middle Level

Q1 Medium

Points

A

B

C

, and

D

lie on a line in that order. Point

B

is the midpoint of segment

\overline{A C}

, and point

C

is the midpoint of segment

\overline{A D}

. If the length of segment

\overline{B D}

is 27, what is the length of segment

\overline{A B}

A 6

B 9

C 12

D 18

E 27

Show Solution

Let the length of segment $\overline{A B}$ be $x$ . Since $B$ is the midpoint of $\overline{A C}$ , the length of $\overline{B C}$ is also $x$ . This makes the total length of $\overline{A C}$ equal to $x + x = 2 x$ .

Since $C$ is the midpoint of $\overline{A D}$ , the length of $\overline{C D}$ must equal the length of $\overline{A C}$ , which is $2 x$ .
The length of $\overline{B D}$ is the sum of the lengths of $\overline{B C}$ and $\overline{C D}$ . So, $\overline{B D} = x + 2 x = 3 x$ .
We are given that the length of $\overline{B D}$ is 27, so $3 x = 27$ . Dividing by 3, we find $x = 9$ .
Therefore, the length of segment $\overline{A B}$ is 9.

Answer: B

Q2 Medium

A straight wooden board is cut into three segments. The first segment is

\frac{1}{4}

of the total length of the board. The second segment is 15 inches long. The third segment is

\frac{1}{3}

of the total length of the board. What was the original length of the board, in inches?

A 24

B 30

C 36

D 48

E 60

Show Solution

Let $x$ be the total original length of the board. The lengths of the three segments add up to the total length: $\frac{1}{4} x + 15 + \frac{1}{3} x = x$ .

First, find a common denominator for the fractions to combine them, which is 12: $\frac{3}{12} x + 15 + \frac{4}{12} x = x$ .
Combine the fractions: $\frac{7}{12} x + 15 = x$ .
Subtract $\frac{7}{12} x$ from both sides to get the $x$ terms together: $15 = x - \frac{7}{12} x$ .
$15 = \frac{5}{12} x$ .
Multiply both sides by the reciprocal, $\frac{12}{5}$ , to solve for $x$ :
$x = 15 \times \frac{12}{5} = 3 \times 12 = 36$ .
The original length of the board was 36 inches.

Answer: C

Q3 Medium

Points

P

Q

R

, and

S

lie on a straight line in that order. The length of segment

\overline{P S}

is 50. If segment

\overline{P Q}

is twice as long as segment

\overline{QR}

, and segment

\overline{R S}

has a length of 14, what is the length of segment

\overline{P Q}

A 12

B 16

C 20

D 24

E 36

Show Solution

The total length of the segment is the sum of its parts: $\overline{P Q} + \overline{QR} + \overline{R S} = \overline{P S}$ .

We are given $\overline{P S} = 50$ and $\overline{R S} = 14$ . Substituting these values into the equation gives: $\overline{P Q} + \overline{QR} + 14 = 50$ .
Subtracting 14 from both sides gives $\overline{P Q} + \overline{QR} = 36$ .
We are also told that $\overline{P Q}$ is twice as long as $\overline{QR}$ . Let the length of $\overline{QR}$ be $x$ . Then the length of $\overline{P Q}$ is $2 x$ .
Substitute these into the equation: $2 x + x = 36$ .
$3 x = 36$ .
Divide by 3 to find $x = 12$ . This is the length of $\overline{QR}$ .
The question asks for the length of $\overline{P Q}$ , which is $2 x$ . So, the length is $2 \times 12 = 24$ .

Answer: D

Tips & Strategies

Draw it out! The SSAT won't always give you a picture. If a question says 'Point B is between A and C', quickly draw a line with those three dots on your scratch paper. It makes the math so much easier to see!
Hunt for the word 'midpoint' or 'bisects'. These are secret SSAT clues that mean 'divide by 2' or 'set the two pieces equal to each other'.

Common Mistakes

Watch out for assuming a point is in the exact middle just because it's 'between' two other points. 'Between' just means it's on the line somewhere. Only assume it's perfectly in the middle if you see the word 'midpoint'!
Don't forget to answer the actual question! Sometimes the SSAT asks you to find the value of $x$ , but other times it asks for the length of a specific segment (like 'What is the length of AB?'). Always double-check what the question wants before picking your answer.

Frequently Asked Questions

What does 'collinear' mean on the SSAT?

'Collinear' is just a fancy geometry word that means 'on the same line'. If points A, B, and C are collinear, you can draw one perfectly straight line connecting all three of them!

Will I need to use a ruler on the SSAT?

Nope! You aren't allowed to bring a ruler to the SSAT. The test questions are designed to be solved using math rules, not by measuring. Plus, the drawings on the test are often 'not drawn to scale', so measuring wouldn't help anyway!

What if a segment length comes out to a fraction?

That's totally okay! Lengths can be fractions or decimals. For example, a piece of string could easily be $5 \frac{1}{2}$ inches long. Just use your normal fraction math rules to solve.

What is a 'bisector'?

Think of a bisector like a karate chop right down the middle! A segment bisector is a point, line, or ray that cuts a line segment into two perfectly equal halves.

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