Calculator Input
Formula Used
If AD bisects angle A in triangle ABC, and D lies on BC, then:
AB / AC = BD / DC
The cross product form is:
AB × DC = AC × BD
- To find DC:
DC = (AC × BD) / AB - To find BD:
BD = (AB × DC) / AC - To find AB:
AB = (AC × BD) / DC - To find AC:
AC = (AB × DC) / BD - Total base:
BC = BD + DC
How to Use This Calculator
Choose the value you want to calculate. Enter the known triangle sides and base segments. Use the same unit for every length. For example, do not mix inches with centimeters. Leave the unknown field empty when using a missing value mode. Press the calculate button. The result appears above the form and below the page header.
You can also use verify mode. In that mode, enter AB, AC, BD, and DC. The tool compares side ratio and segment ratio. It also checks both cross products. A small difference may appear because of decimal rounding.
Example Data Table
| Case | AB | AC | BD | DC | BC | Result |
|---|---|---|---|---|---|---|
| Find DC | 12 | 18 | 8 | 12 | 20 | 12 / 18 = 8 / 12 |
| Find BD | 10 | 15 | 6 | 9 | 15 | 10 / 15 = 6 / 9 |
| Verify | 7 | 14 | 5 | 10 | 15 | Theorem satisfied |
Angle Bisector Theorem Guide
What the Theorem Means
The angle bisector theorem is a useful triangle rule. It connects a divided angle with a divided opposite side. Suppose a line starts at vertex A. It cuts angle A into two equal angles. It then meets side BC at point D. The theorem says the nearby sides decide the split on BC.
Why Ratios Matter
The main idea is ratio balance. Side AB is compared with side AC. Segment BD is compared with segment DC. When AD is a true angle bisector, these two ratios are equal. This makes missing side problems easier. You can solve them with one proportion.
Practical Learning Value
This calculator helps students check homework. It also helps teachers prepare examples. Geometry problems often give three values. The fourth value must be found. Manual solving can be slow when decimals appear. This tool gives quick results and shows cross product checks.
Using the Output
The result panel gives the missing value, total base length, ratios, and verification data. The chart compares side lengths and base segments. The CSV option is useful for spreadsheets. The PDF option is useful for saving work or sharing a solution.
Accuracy Tips
Always use positive lengths. Keep all values in one unit. Round only at the final step. If a verification result shows a tiny difference, it may come from decimal rounding. A large difference means the line may not be an exact angle bisector.
Where It Is Used
This theorem appears in geometry, trigonometry, drafting, design, and construction layouts. It is also useful in proofs. Once the ratio relation is clear, many triangle questions become direct and simple.
FAQs
1. What is the angle bisector theorem?
It states that an angle bisector in a triangle divides the opposite side in the same ratio as the two adjacent sides.
2. What values can this calculator find?
It can find AB, AC, BD, or DC when the other required three values are known.
3. Can I use decimals?
Yes. Decimal lengths are allowed. Keep enough decimal places for better accuracy and round only after the final result.
4. What does verify mode do?
Verify mode compares AB divided by AC with BD divided by DC. It also checks the cross products.
5. Why must values be positive?
Triangle side lengths and segment lengths represent distances. Distance values cannot be zero or negative in this theorem.
6. What is BC in the result?
BC is the whole opposite side. It is calculated by adding segment BD and segment DC.
7. Does this prove AD is an angle bisector?
If the ratios match, the values satisfy the theorem. A diagram or given geometry may still be needed for a formal proof.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a printable calculation summary.